[[Attachments/Optics Derived .docx|Optics Derived ]] [[Optics Angular Resolution or Earth Curve]] The mainstream explanation for bottom-up obstruction is that it is caused by the geometric curvature of the Earth. However, this effect can be entirely explained by optical phenomena related to angular resolution and perspective. The loss of information at the horizon can be modeled using the values of optical limits of perception, challenging the notion that it is due to physical curvature. The phenomenon of objects disappearing from the bottom up as they move away from an observer can be explained through the concepts of perspective, angular resolution, optical illusions, and the refraction of light. ## Angular Resolution and Perspective Angular resolution is the smallest angle at which two points can be distinguished as separate by the human eye. This limit is crucial in understanding how we perceive distant objects. As an observer lowers their height, the horizon line appears closer, which affects the compression and expansion rate of angles. This loss of information is a function of angular resolution and perspective, not curvature. ## Effect of Observer Height The observer's height has a direct impact on the perception of objects. When the observer's height is lowered, the angles close to the ground compress, making the horizon appear closer. Conversely, the angles above the eyeline expand. This compression at the bottom and expansion at the top alter how objects are visually processed, leading to the bottom-up disappearance of objects as they move away. ## Proportional Relationships The relationship between the observer's height and the movement of the lower horizon line is proportional. The change in the size of objects within a fixed perspective is linear. Objects maintain their relative proportions above and below the eyeline throughout the entire observation. ## Optical Phenomena and Refraction Atmospheric conditions, such as refraction and aberrations, also play a role in altering the perceived size and shape of distant objects. These optical phenomena can cause objects to appear distorted or to disappear at certain angles, contributing to the bottom-up obstruction effect. ## Visual Compression and Expansion Angles of sight compress or expand based on the observer's height. As an observer lowers their height, the angles close to the ground compress, making the horizon appear closer. Changing the observer's height affects the angles of sight, compressing them more at the bottom (ground level) than at the top (sky level). This alteration results in a change in how information is visually processed and how far one can see. Why do things disappear bottom up? Lets focus on the concept of perspective and optical phenomena. For this, we delve into topics such as angular resolution, optical illusions, and the refraction of light, and the angular resolution limit. the effect of lowering the observer's height on the perception of objects in a fixed perspective. the loss of information is a function of angular resolution and perspective, not curvature. the observer's movement vertically and the distance of the object are related. When the observer's height is lowered, the horizon line is brought closer, affecting the compression and expansion rate of angles. objects maintain their relative proportions above and below the eyeline throughout the entire observation. the relationship between the observer's height and the movement of the lower horizon line is a proportional relationship the relationship between the distance and the change in the size of objects within a fixed perspective is a linear relationship Perspective, observer height, the diffraction limit, the attenuation of light, and angular resolution limit all contribute to the visual phenomenon of objects appearing to disappear bottom-up at a distance. The eye's resolution angle, which is the minimum angle at which two points can be distinguished as separate, plays also plays a crucial role in the perception of objects at a distance. For example involving two equal-sized balls at a given distance [pictured below], the one above the observer's line of sight appears smaller as it moves away, while the one below appears larger. They also illustrate that there is a specific angle at which objects disappear from sight, referred to as Θ (theta), and this angle is reached sooner for objects below the eye line. The images below illustrate how angles of sight compress or expand based on the observer's height and how objects of different heights are perceived by the human eye. They convey that as an observer lowers their height, the angles close to the ground compress, making the horizon appear closer. Changing the observer's height affects the angles of sight, compressing them more at the bottom (ground level) than at the top (sky level). This alteration results in a change in how information is visually processed and how far one can see. Conversely, the angles above expand. An object traveling at a higher altitude would show little change in angular size before it sets into the lower horizon. Although the total number of potential angles of vision above and below the eye line is the same, the compression rate of these angles differs. This variance in compression affects how objects are visually perceived as they move away from the observer. Objects can seem to disappear or "set" into the horizon or other larger objects (like buildings) when viewed from certain angles. When one looks up at a very steep angle, the floors towards the top of a building compress visually, making it harder to discern details or even see the highest floors. This compression of angles leads to a loss of information and contributes to the "disappearance" of objects that are not actually obscured by any physical barriers. Also, there are of course atmospheric conditions such as aberrations and refraction that can alter the perceived size and shape of distant objects. The higher the ceiling, the steeper the angular compression, and the faster the descent rate to the vanishing line. This is EXACTLY how the sun and moon appear to set, maintaining a relatively constant size due to their high upper boundary and short travel distance within the observer's perspective before they meet the horizon line. For more info: Bottom up obstruction on a level plane https://publish.obsidian.md/shanesql/Bottom+up+obstruction+on+a+level+plane Optics Angular Resolution or Earth Curve https://publish.obsidian.md/shanesql/Optics+Angular+Resolution+or+Earth+Curve ![[Attachments/Pasted image 20240827112001.png]] Lets examine the phenomenon whereby objects disappear from the bottom up, attributing this to optical and physical factors. This disappearance is linked to the exponential divergence of light, subsequently converged by the lens of the eye to create a perspective. It is noted that when an observer lowers their position, the angles at which information at the bottom is received compress, drawing the lower horizon line nearer. ![[Attachments/vlcsnap-00662 1.png]] Taller objects, those extending above the observer's eye line, retain their proportions yet seem to descend more rapidly. The rate at which different parts of an object appear to descend is also influenced by their position relative to the observer's eye line. Changes in the observer's height alter the distance to the horizon in a linear manner. Despite maintaining their relative proportions, the lower parts of objects compress at a rate distinct from their upper parts, making them seem to merge into the lower horizon line. This phenomenon results from perspective and angular compression rather than curvature. Optical principles, specifically, diffraction and refraction effects, cause objects to seem to vanish from the bottom up over great distances. This is due to angular resolution limit, diffraction, and refraction. The angular resolution limit is identified as the point where converging rays from an image or object meet at the observer's specific vanishing point, facilitating the bottom-up disappearance. Refraction and diffraction critically influence the perception of objects across vast distances, elucidating the complex interplay of optical phenomena involved. There are various optical phenomena and potential reasons for the bottom-up disappearance of objects. The effects of the diffraction limit and warm air above water are examined as factors contributing to this phenomenon. The concept of perspective and the effect of upward refraction on the visibility of distant objects are explored. Additionally, the role of the optics of the human eye is integral in perceiving distant objects. Angular Resolution and Visibility the vanishing line and why things disappear bottom up. the angular resolution limit affects the visibility of objects. As an object moves further away, the angle of resolution across gets smaller, reaching a limit at about 0.02 to 0.03 degrees. At this point, the cones in the retina can no longer resolve the image, and the information cannot be distinguished. Refraction and Mirage Effects how inferior mirages occur when light from an object meets a critical angle of refraction and reflects or refracts away from the observer's eye. The impact of atmospheric conditions on refraction, resulting in varying lensing effects and distortion near sea level. It highlights the complex variations in lower-level refraction, causing the objects to appear higher or lower, and even leading to inversion effects. Specific time-lapse video footage is referenced to discuss the importance of the angular resolution limit and upward refraction in observing distant objects, particularly the Chicago skyline from across Lake Michigan. ![[Attachments/Pasted image 20240711214032.png]] ![[Attachments/Pasted image 20240711214028.png]] ![[Attachments/vlcsnap-00545.png]] ![[Attachments/vlcsnap-00547.png]] ![[Attachments/vlcsnap-00548.png]] ![[Attachments/vlcsnap-00525.png]] ![[Attachments/vlcsnap-00544.png]] ![[Attachments/vlcsnap-00549.png]] ![[Attachments/vlcsnap-00583.png]] ![[Attachments/vlcsnap-00584.png]] ![[Attachments/vlcsnap-00586.png]] ### The sun ![[Attachments/vlcsnap-00587.png]] ![[Attachments/vlcsnap-00588.png]] ![[Attachments/vlcsnap-00589.png]] ![[Attachments/vlcsnap-00591.png]] ![[Attachments/vlcsnap-00597.png]] ![[Attachments/vlcsnap-00599.png]] ![[Attachments/vlcsnap-00600.png]] ![[Attachments/vlcsnap-00603.png]] ![[Attachments/vlcsnap-00604.png]] ![[Attachments/vlcsnap-00614.png]] ![[Attachments/vlcsnap-00631.png]] ![[Attachments/vlcsnap-00633.png]] ![[Attachments/vlcsnap-00651.png]] ![[Attachments/vlcsnap-00652.png]] ![[Attachments/vlcsnap-00653.png]] ![[Attachments/vlcsnap-00654.png]] ![[Attachments/vlcsnap-00661.png]] ![[Attachments/vlcsnap-00662.png]] _________________________________________________________________________________ ## AI Summary of the video: Key Points 1. The paper thoroughly investigates the disappearance of people and objects at the horizon, exploring whether the loss of information is due to optical or physical factors. It clarifies that the disappearance is primarily attributed to optical phenomena, specifically the angular resolution limit of the observer's eye, and refutes the idea that it is solely due to the physical curvature of the Earth. [ 1 ] 2. It provides an in-depth analysis of optical factors and time-lapse footage, presenting evidence that challenges the traditional understanding of the disappearance of distant objects. This includes the discussion of upward refraction phenomena and examples of time-lapse footage showing the visibility of buildings despite the Earth's curvature, contradicting the expected curvature-based obstruction. [ 3536 ] 3. The paper also presents a challenge to the heliocentric model and traditional scientific theories related to the Earth’s orbit, gravity, and rotation. It suggests that modern scientific concepts are linked to religious and occult ideologies and urges re-examination of cosmological beliefs based on biblical teachings and alternative philosophical interpretations. [ 4849 ] Synopsis The research paper discusses the phenomenon of people disappearing at the horizon and explores whether the loss of information at the horizon is attributed to optical or physical factors. It explains that if the loss of information at the horizon is physical, people disappear by walking over the curve, but if it's optical, people disappear due to the angular resolution limit of the observer. The paper provides detailed explanations and diagrams of how people and objects disappear from perspective based on the observer's eye level and distance. It clarifies the concept that the loss of information at the horizon is a result of perspective and angular resolution rather than due to a physical curvature. The paper also discusses the phenomenon of objects compressing or expanding relative to each other as they move further away from the observer, and how the observer's height can influence the visual perception and the convergence of the information. It emphasizes that the rate at which objects diminish in size within a fixed perspective is linear and illustrates this phenomenon by using examples of objects in a fixed perspective. The paper delves into the concept of lens convergence, the proportional compression of light rays, and the angular resolution limit of the human eye. It concludes by highlighting the direct proportionality between the amount of lowering the observer's height and the corresponding closer horizon line and providing a detailed explanation of how objects above and below the eye line maintain their relative positions and proportions. Overall, the paper provides a comprehensive and detailed analysis of the phenomenon of people and objects disappearing from perspective and clarifies the optical and physical factors influencing these visual phenomena. [ 1 ] The Angular Resolution Limit and Diffraction The research paper discusses the phenomenon of people disappearing at the horizon, focusing on whether the loss of information at the horizon is attributed to optical or physical factors. The paper explores the ways in which people disappear from perspective and provides a detailed analysis of the angular resolution limit and diffraction limit. The paper discusses how the visual acuity of the average person is taken from the central portion of the back of the retina, where rods and cones are most concentrated in the fovea. It also delves into the formula for the angular resolution limit and its relationship to the wavelength of light, diameter of the aperture, and the constant 1.22. The paper provides a mathematical understanding of the angular resolution limit, discussing the first-order Bessel function and its application to the diffraction limit. It explains diffraction patterns of light scattering through an aperture or a pinhole and the airy pattern, detailing the bright disks and dark fringes. The paper also discusses the phenomenon of mirages and refraction effects, as well as the impact of temperature and air currents on refraction. It explains how the warm air below and the colder air above create upward refraction of objects and bring them downward from the top, leading to the appearance of buildings as lower than they actually are. The paper provides a comprehensive investigation into the complex interplay of optical and physical factors that contribute to the phenomenon of people disappearing at the horizon. [ 141 ] Optical Factors and Time-Lapse Footage The paper makes several key points regarding the phenomenon of people disappearing at the horizon. The author discusses how the loss of information at the horizon is attributed to optical factors rather than physical curvature. The paper explains that the disappearance of distant objects, such as buildings and ships, is a result of a combination of angular resolution limits and upward refraction phenomena. The author presents various examples, including time-lapse footage of the Chicago skyline taken from different distances and locations. The footage shows that the buildings remain visible despite the curvature of the Earth, challenging the traditional understanding of how objects disappear from view at a distance. [ 1 ] The paper delves into the concept of refraction, using historical experiments such as Airy's failure to illustrate how light behaves in the atmosphere. The author emphasizes the optical nature of the phenomenon, refuting the traditional explanation that objects disappear behind the curvature of the Earth. The paper argues that the visibility of distant objects, as observed in various time-lapse footage, contradicts the expected curvature-based obstruction. [ 37 ] A Challenge to Heliocentrism and Scientific Theories Furthermore, the paper explores the idea that the Earth is motionless and flat, challenging the traditional heliocentric model. The author argues that the current scientific understanding of gravity, Earth's rotation, and its orbit around the sun lacks evidence. The paper links these scientific concepts with religious and occult ideologies, suggesting that modern scientific theories are founded on a mystery religion. It emphasizes the importance of seeking truth and God's guidance, urging readers to re-examine their understanding of the world. [ 4849 ] The images explain the concept of perspective, angular resolution, and how these factors affect the visibility of objects at varying distances and heights. They provide visual demonstrations of how objects can seem to disappear or "set" into the horizon or other larger objects (like buildings) when viewed from certain angles. One image uses tall buildings as an example to explain that when one looks up at a very steep angle, the floors towards the top of a building compress visually, making it harder to discern details or even see the highest floors. This compression of angles leads to a loss of information and contributes to the "disappearance" of objects that are not actually obscured by any physical barriers. The images also clarify the terms used in perspective drawing such as orthogonal lines, transversal lines, and vanishing points. They illustrate how the eye's resolution angle, which is the minimum angle at which two points can be distinguished as separate, plays a crucial role in the perception of objects at a distance. In addition, the images discuss how changing the observer's height affects the angles of sight, compressing them more at the bottom (ground level) than at the top (sky level). This alteration results in a change in how information is visually processed and how far one can see. Lastly, the visuals demonstrate how objects at a lower height have their bottom parts vanish from sight first as the distance increases, using a real-life example of a bottle filmed from different heights on a street to illustrate the point. Overall, these images aim to provide a comprehensive explanation of the visual mechanics behind the disappearing bottoms of objects at a distance due to perspective and the eye's angular resolution capabilities. ![[Attachments/vlcsnap-00547 1.png]] ![[Attachments/vlcsnap-00548 1.png]] ![[Attachments/vlcsnap-00549 1.png]] ![[Attachments/vlcsnap-00525 1.png]] ![[Attachments/vlcsnap-00544 1.png]] ![[Attachments/vlcsnap-00545 1.png]] ![[Attachments/Pasted image 20240626094833.png]] compare the scaling of squares to their limit of compression on the ceiling versus floor this is the most obvious way i can think to demonstrate how those angles of propagation from the surface compress at a faster rate with distance, than the angles of propagation from the ceiling ![[Attachments/Pasted image 20240626094847.png]] if we remove the floor then the propagation angles are the same from the ceiling to the other ceiling below ![[Attachments/Pasted image 20240626094910.png]] without reference points in the sky, we don’t know how to comprehend the scaling rates of image compresssion from the angles of propagation of light that reflects or is produced by enormous objects at enormous height, when then move away or toward us (edited) ![[Attachments/Pasted image 20240626094923.png]] ![[Attachments/Pasted image 20240626095140.png]] ## Full transcript from the video Angular Resolution and our world. **Angular Resolution and Our World** I will lay out the issue from the outset. The loss of information at our horizon is either an optical phenomenon or a physical one. If it is physical, people disappear by walking over the curve. If it's optical, people disappear by the angular resolution limit of the observer. Option one the ground is brought to your eye level. The person disappears from perspective by walking straight away. Vanishing bottom first from things getting smaller with distance and and angular resolution. Limit.  As you can see in the diagram, the eye line stays level, the object is getting smaller with distance and the horizon is brought to the eye level. Uh, designated with our. A yellow and purple lines. Option two the eye is brought to ground level. The vanishing person is represented orthographically without regard to perspective, and your focus continually shifts downward to observe them going over a curved surface and vanishing at the same rate as the angular resolution limit of the human eye. Within the diagram, you can see that the horizon is now brought downward to ground level. The eye line shifts to observe an object physically going over curvature. Perspective effects and angular resolution are ignored. After reviewing the above options, we will see how the optical angular resolution limit of the human eye was replaced by the physical curvature of the Earth, and we will then use the angular resolution limit to calculate a sphere with a radius of 3959 miles, two 3978 miles. So option one the ground is brought to your eye level. The person disappears from perspective by walking straight away, vanishing bottom first from things getting smaller with distance and angular resolution limit.  We will start with our lens, our eye looking at a block. Two light rays are sent to our eye. We know that we see the world through spherical geometry due to the shape of our eye. However, even at a relatively short distance away from our eye, the spherical geometry assumes planar representations. So in that respect, the use of planar geometry is accurate for illustrative purposes. And if you'd like to learn more about that, please see the video links in the description below. We can add another object and create an alternating pattern of any kind or substance. We can assume for the moment that this is orange and blue paint on a wall. This side on view is termed orthographic. We can add a floor and a ceiling.  If we shorten the distance to the object, the angles at which they are seen increase outward. So the yellow angles are wider than the green and the green are wider than the red. Each of these blocks represents a ten foot increment. If we zoom in, you can see that the closer object now has more divergent visual field lines than the more distant object. Note that these are parallel lines. When we render this to our perspective view, what we see may not represent this orthographic reality, and these lines will not appear parallel, they will converge. So keep in mind that the floor and ceiling run parallel, as well as any other line along the side that is parallel to the floor or ceiling. Again, these are ten foot spacings between each column of paint.   We can also progressively shorten the distance to the object again. We've now put multiple rows of paint along our corridor, six of them, for example, and we can move them progressively closer or farther. And you see that the angle of light is more and more divergent as it gets closer, just like we did for adding the nearby pink columns earlier.  So the information at the periphery changes at a faster rate than at the center. But our lens converges these rays at a rate proportional to their exponential divergence. And we view a linear world. If you reverse the information that is being converged by the eye. You start to see in general terms what light the lens is converging, and in turn what the visual cortex interprets. And this becomes our perspective. So again in the top view here, this is the light rays extending off that we are converging through our lens in order to converge them at a focal point behind, uh, our uh, behind our lens and then focus that onto our retina. Uh, you can see if we then just reverse these lines as the information, uh, coming off in or being reflected off of the object by a light source, either of the sun or some kind of artificial light source exponentially. You see that? That is what we are trying to converge again. So the object itself is linear. We see it as linear, but the information that is spread between the two points is exponential exponentially, given off by the light source exponentially, then converge by our lens system.  So keep in mind we we see a linear world. Um, and the linear world exists as it is. It is linear. Uh, it is not an exponential. Um, so again, you can be looking side on view, uh, here with these diagrams, with this being your lens of the eye. Uh, within the, the head of a person. Or you could be looking top down as if you were looking down on the top of their head. And these are the left and right walls instead of the ceiling and the floor. Either way, all things are parallel and perpendicular to one another, uh, within the actual real world. But the light rays that are being extended off the object are exponentially radiating outward from a point source according to the inverse square law, uh, which means that, uh, as the distance increases, uh, or the object increases at a rate, uh, the inversely proportional to the square of the distance. So, um, again, these oncoming light rays are what we converge to form that perspective. And we're going to cover that in more detail as we go. Uh, but this is kind of a cursory look at our perspective view. We'll, um, cover what these, um, lines represent as we go. There are transversal lines within the these are the rectangular portions of this view. And these the light rays that are coming off over here, the red and green and yellow radiating, uh, light rays essentially become, um, somewhat of orthogonal lines, but we'll cover that as we go. So again, if an outer ray is spreading out more and hits the lens at a greater angle, it's then refocused by the proportional amount. Uh, and the image remains linear by the lens. So image size, which is height or h, becomes inversely proportional to the distance away. So again height proportional to one over the distance. So if we double the distance the size reduces by one half. And so on. That is how we view the world. Uh so as we'll see with distance we focus light across a larger focal length and limited aperture. So we limit the rays with which we are viewing the object. And there is an angle at which they become so close that they are interpreted as a single ray for the human eye. This angle is about one arc minute, or 0.02 to 0.03 degrees, and it represents the vanishing point of our vision. Since we must have two light rays or two light sources to construct an image. We will cover this in detail as we go forward, but for now, just realize that we can't see forever. There is a vanishing point where information is simply too compressed. And so I'm just trying to show you kind of a very basic illustration of what is going on. We have an object that is, uh, sending out information, um, either reflected light or uh, or light, uh, itself. Um, it's sending it out exponentially as a point source to our eye. Our eye is exponentially converging that information. Uh, and we interpret that as our perspective. For illustrative purposes, we can view this another way when we rotate the floor and ceiling until they meet along the center line or eyeline, the point that they meet along this vanishing line is the vanishing point. Recall that while a near column may appear larger than a further column, they both are the same and ten feet is still ten feet, even though one is bigger than the other, potentially. So again we are having the distance and doubling the size. Uh, height equal to or proportional to the inverse of the distance. And so ten feet as it gets closer looks larger than ten feet. That is further aw So objects linearly scale inversely proportional to the distance, like we said. To prove this to yourself, measure out the distance in a space like a hallway or a corridor. Take a series of pictures at evenly spaced distances to an object, and you will see that as you double the distance, the size decreases by 50%. Double it again and it reduces another 50%, and so on. So again, double the distance half the size. This 80 inch door changes apparent size with distance. It gets bigger as its closer and its smaller as it gets further away. So at 120in away, its a certain size within our visual field. At 240in away, its half the size and it halves again at 480in away. Again, we can rewrite our formula to compare objects as a ratio.  So with our height is inversely proportional to the distance, we can rewrite as height one over height two is equal to distance two over distance one. That's just comparing to, um, two object heights with two distances to one another. So the 80 inch door is half the size at twice the distance, or one third the size at three times the distance, and so forth. But the door is always itself 80in. It's just perceived differently. This may seem like a trivial point, but it's important.  So in general terms, we can now translate this model to a one point perspective to see how the vanishing line or eyeline creates the vanishing point. It is the limit to our visual world. Recall that angles expand exponentially by the inverse square law as they near you, and they are exponentially converged again by the lens so as to retain their same proportions. If they depart at a smaller angle due to a far distance, uh, they are converged at a smaller angle and appear smaller. So we will now go into some detail on the vanishing point. We see through a convex lens system, where light is inverted on our retina and transmitted through our optic nerves, and optic radiation inverted in an inverted form, and are reversed again in our visual cortex to appear as if upright. So I'll discuss a convex lens system briefly, then apply it to our vision. Light bends through a prism via a process called refraction. It's a velocity change of light within another medium, such that the angle at which it travels is altered. The higher the refractive index of the medium, the slower the light travels and the sharper the angle the ray takes as it travels through it. If we make our arrows, some kayakers, uh, who connected themselves with a metal bar, we can see what is happening as they head toward the more refractive medium. Like it were, uh, thick sludge. So on the left here, we have basically our lens or our prism here, which we won't call it a lens just yet, but it's a it's a prism, uh, where the light enters, it hits this angle of this medium. It's traveling in air here. It'll be glass, let's say here, uh, it hits the medium. The light on this, uh, downward, uh, edge strikes first and it gets bent. It bends the ray downward. Uh, once the ray is closer to exiting the surface here on this, along this angle. Now, the the, uh, the the the top portion hits the air, uh, soonest, and the they start to go faster, essentially. And, um, while the bottom part is remaining to go slow and this light then gets bent downward. So looking down on the kayakers, the ones on the bottom hit the sludge first and go slower as the ones on the top continue on like we talked about. So the line of them is bent toward a new downward path, or whichever vector you please. Uh, left to right. So on and so forth. Looking top down, looking side on. Either way. If we invert the prism, the opposite occurs. We can. So in other words, we're just flipping the prism upside down. We can add them in such a way these two prisms together as to create a lens and a lens, such as the one in our eye or in your camera will converge all light, no matter its entry point to a focal point. There will be a point at which these two, uh, similar prisms will converge the light to a single point. Um, this lens is a convex lens on each side. So we can create another focal point on the other side at the same distance from the lens. So in other words, light travels through from left to right. And it's converged over here on the right side to a focal point. But equally the light can travel through, uh, going right to left and do the same thing on the other side as a focal point on the left. So we can simplify our prisms to to make it more of a standard lens, what we're used to and draw our focal points f one at a given distance from our lens, distance one or d one, and doubled that distance for a second focal point f two at a distance d two. So that's on one side of the lens. And we'll call on the other side of the lens f two prime and f one prime. And they represent the focal points on the other side of this biconvex lens. So we need two incident rays to create an image. If we put an object in front of the lens, the rays of light must enter by their respective focal points. Uh, or an image is not converged or properly resolved on the other side. So for simplicity, the first ray off the top of the tree goes through f one prime, and the second ray off the top of the tree goes through f one. Where they cross is where a real image is formed, and it is always inverted. Because it is a convex lens, the visual cortex reverses the image in order for us to perceive the image right side up. And if there are any questions concerning the focal points and lens systems, there are many videos online that go into depth on that system. But not not to get into the physics of lenses too much, but the image is inverted and smaller on the other side of the lens. If beyond f two, then for our purposes, that's what we're going to focus on. If I pull back farther beyond f two, it gets smaller and smaller. Again, we're not going to discuss lens problems where the lens is concave or the image is inside f two, because we see the world at a distance beyond f two, F one prime represents about 1.6cm within our eye. So as long as you're focusing beyond about 3.2cm f two prime in front of your eye, the world is viewed as depicted. We can now place our lens diagram or our diagram within the optic system. The lens is the lens of our eye, or the lens of a camera, and the f one behind the lens is where the light rays converge. The rays then diverge and hit the retina or camera sensor with an inverted image, and that is again smaller with distance. So you can see our two point sources coming off this. Let's say this is our up here in this diagram. This is our tree, the top of the tree that's going to be focused to the uh, to the back of our retina. The other ray from that same, uh, source is sent off, travels through f one prime, and then travels through the lens, equally converging, uh, to create an image in the back of our retina. Um, so again, if you move the object further out, which is what we're depicting with the second tree, uh, the you can just see that the object gets smaller and smaller, uh, within what is projected to the back of the retina. This blue line here just represents a, uh, a central ray, essentially, uh, split between the two. Uh, and it is the ray that is neither, uh, diverged nor converged, uh, along the lens system. But as you can see, with further distance, you can kind of conceptualize how this Blu ray would get closer and closer and closer to this. Uh, Ray going through f one prime. And at a certain point, those rays will become indistinguishable from one another. And in the same way, there becomes a point where the light that is hitting our eye is, uh, so close together that the information is just not resolved. So again, we can now, um, move on hopefully a little bit with our lens diagram. We do have some ability to manipulate this incoming light, so it's not a just pure lens system. Um, there are three main ways that we focus near and far. The first is the convergence of the eyes themselves via the extraocular muscles. The second is the stretch and relaxation of the lens via the ciliary muscles, and the third is the constriction and dilation of our pupil or aperture via the iris. The relaxation of the ciliary muscles make the lens more convex, and as it bulges out, the pupil dilates. The contraction of the ciliary muscles makes the lens less convex, and it flattens and the pupil constricts. This is the Helmholtz theory, and there are other theories for how exactly this occurs. But regardless, when taken together, the three effects represent the accommodation triad for near and far vision. So near vision is characterized by a more convex lens and a dilated pupil, and far vision is characterized by a less convex lens and a constricted pupil. This is called the accommodation triad. Again, here's your extraocular muscles. There are several of them that attach and fix the eye and allow the eye to converge an image, and the ciliary muscles are attached to the lens, and pull the lens and distort it to make it more convergent or less convergent, with being more convex and less convex, and then by default the pupil dilates or constricts based on the convergence, um, or bulging of the lens. Uh, but again, we can't accommodate forever. Uh, our lens system does reach its maximum. So going back to our diagram again with the distance beyond f two prime, a real image from converge. Light is formed on the sensor or the back of our retina. As the image extends beyond f two prime, there is less refraction occurring of these incident rays. So again we'll take our blue central ray. Again. Just to reiterate the point as an illustration, the central light ray is not refracted as it passes through a convex lens. So as we go far out beyond f two prime, you can see how the central ray will approach the ray through f one prime. As we'll see, image integrity starts to break down and light rays are viewed as indistinct. The lens convergence is at its minimum, and we reach a limitation on the system based on the manner in which it entered. And that is by diffraction. So as we'll see, the pupil diffracts the light and the lens convergences. So when the lens can no longer converge the image, our image will remain subject to something called diffraction and will reach a diffraction limit or angular resolution limit. So again, so I'm kind of trying to illustrate that here below as these rays merge, um, and the angle gets smaller with distance our vision becomes diffraction limited and a horizon line forms. Uh, this occurs at an angular resolution limit angle, which we will discuss. And for all angles less than that angle until the horizontal eye line, uh, theoretical zero angle, uh, would be produced. So anything at that particular point where the light rays are so merged together or so close to one another that you cannot resolve them, that angle plus any angle less than it up to the zero degree point of our eye line is not visible. So with this in mind, let's go into some detail on how we see the world. Again. We'll point to our, uh, perspective, uh, uh, illustrations here. Um, up to this point, we've kind of been equidistant from our ceiling, floor or walls. We've been right in the middle. Objects on either side of the vanishing line compress and expand relative to one another at the same rate. So these ten foot walls and ceilings all, uh, went towards our vanishing point at the same rate. Um, if my eye is in the middle of the corridor, the walls, ceiling, and floor all compress and expand linearly and equally proportioned as I move near and far. So likewise, a near object in the middle of my eye line remains in the middle as it moves away, and it diminishes in size proportionally in all directions. So if I put something right in front of my eye and it moves straight away, it will get smaller and smaller and smaller in proportion. Uh, according to these essentially, uh, rectangles, which will are called transversal lines. And this is how we model a 3D world into 2D. So the vanishing line is the point at which information is lost due to too small and angle. This point is always perpendicular to the back of our retina. The horizon is always at eye level because it is our eye line. It is the point where the resolution angle is so small we perceive it and all angles less than it as a zero degree central angle or a straight eye line. This angle can occur 360 degrees around our eyeline like a cone, but for simplicity we will model it above and below the eye line. But again, you can look at it from the top down from a 45 degree angle, from a 90 degree angle, from a 270 degree angle is still going to be the same, uh, angle. But for simplicity, we're going to be looking at it as if you're looking at it from the side of the head. So these angles and any information within them represents our horizon. The horizon is the midpoint of our vision. So anything below the eyeline, like a floor, is always the lower half of our visual field. And anything above our eyeline, like a ceiling, is always the upper half. Same for the walls to the either side of us. They're always the left or right fields. Now if we had them. So if the ceiling were tremendously high, as we ascend in height, our personal horizon is always the point at which the top half of our vision meets the bottom half, or commonly where the sky meets the ground. If we were to rise in height, and if the sky and the ground remain in the middle of our eyeline throughout our ascent, the ceiling and floor have remained essentially parallel to one another throughout that time. And as you can see when photographing the horizon and using the same reference point, the horizon does remain constant from 300ft to 30,000ft. So you can take shots up in an airplane and you can reference your camera, your phone to the same exact spot, uh, taking. And you have to account for the change in pitch, uh, uh, of the plane and the ascent angle and so forth. But when you do that, you see that the horizon remains in the middle of your eyeline. Uh, throughout. So as you are taking off and your 300ft up, uh, you will look out the window at a certain angle with a certain distance away from the window and taking reference points along the, uh, let's say the wing or some other thing to make sure that you're taking the exact same, uh, point, and you will see that it remains the same from 300ft to 30,000ft and then back down again. Sometimes there'll be some lensing effects that you have to take into account that you are looking through a convex lens, through an airplane, but you have to keep that in mind when you're looking at these. But if you take the exact same photo, uh, in the exact same position, uh, you will note that the horizon, uh, is where it was when you were low along the ground. Uh, this also may seem like a trivial point, but this will become, uh, important as we go. So we should do some quick definitions before moving on within a one point perspective view, which is how we see the world. The radial lines are orthogonal lines and the rectangular lines are transversal lines, and they are 90 degrees to one another when viewed orthographically for from that side on view. Objects placed in our field of view line up according to these lines. The horizon again is our eye line. It's where the bottom 50% of our vision meets the top 50%, or where the ceiling meets the floor or the sky meets the ground, etc. it is also the vanishing line, and where it crosses our vertical orthogonal line is the vanishing point. It is always 90 degrees to our eye and comprised of all the information lost within a resolution angle of one arc minute. For the human eye, this angle is situated plus or minus uh, from zero degrees from our eye line. So again, if you need to pause and just kind of look at the diagram and try to figure out where these points are within your perspective versus where they are potentially along an orthogonal or orthographic view, then I encourage you to do so. Uh, it's not entirely intuitive. Uh, sometimes. So to see how we have a horizon or vanishing line, as opposed to sometimes seeing a vanishing point, we need to change our original corridor a bit in order for this to become a little more apparent. So keep in mind that these concepts are not mutually exclusive. Exclusive. A vanishing point is present when there is a vanishing horizontal line or horizon, and a horizontal vanishing line is present when you are focused to a vanishing point. But it may not be apparent in the presence of tangible objects to your left or right. So consider that we are now looking down on a person's field of view axially. Uh. We then alter the left wall by rotating it out and away from us. Your view essentially would change from this, which is our side on view and our perspective side. Uh, our perspective one point perspective view. So again, the how we see that top figure is interpreted on the bottom. To this, where we rotate our left wall out a little bit, and your perspective would then change to something like this, which is on the bottom. That left wall is now being rotated outward and away from us, and you can see how the distortion starts to be viewed. And you can model this for yourself just using rectangles and some lines. It's fairly straightforward. So we can continue rotating the left wall outward until perpendicular to the original wall. So we're just going to continue rotating this wall outward and outward and outward. And we're going to see what happens to our perspective view. So the lines denoting the left, upper and lower corners now converge to a single line as the wall rotates, rotates out away from us. As it rotates, it gets further away, and eventually it will be the same angular size as the furthest objects within our perspective view, and it vanishes, leaving the floor and ceiling visible. So again, we're just rotating the, uh, the wall, uh, more and more outward away from us. And eventually as it rotates out, it will get so far away that it will be the same distance, essentially, as that vanishing point is in that original one perspective view where we were equidistant from the walls and ceiling. So if we now rotate the right wall 90 degrees away, we are left with a floor and a ceiling, and the left and right walls compressed into the horizontal vanishing line. So we're going to do the same thing now for the right side. And you can see that if you were to look at this you would see essentially, uh, just a ceiling and a floor if there were no tangible objects, uh, in your view. Um. We. So let's do this now for instead of a left wall, let's remove the ceiling. So I've just rotated the view I've replaced the walls. Um, or you know, I've put the walls back in and we'll just rotate the previous model to. Now we're rotating the ceiling outward and away from us. Now the ceiling is rotated 90 degrees outward. Uh, and it has pivoted away to become a essentially a, a horizontal or in this case, a vertical line. So this is what we would see. There is no visible ceiling, but we still have two walls and a floor in this view. Just think of yourself in an extremely long corridor, and the ceiling is so high that it's outside your visual, uh, your visual field. Um, and we can now do the same thing that we did with the walls. We're just going to rotate the floor now out and away from us. So we're now like, suspended on something, we'll say. And we're rotating the floor away from us. Now you're left with the walls and it's essentially a vertical horizon. You can think of it. Um, or you could say a Verizon, but I thought that was kind of, uh, you know, there's a phone company called Verizon, I guess. But anyway, it's a vertical vertical vanishing line, essentially. So to clarify, if there were no objects on the walls and only perfect paint, this is what you would see. And as soon as we place anything there, though, you would perceive it according to that central vanishing point. Again, that point that is along your eye line would then make itself known by any physical or tangible objects that you place within your view. That will become apparent here as we go. So since massive walls are a rare sight, we'll focus on the horizontal plane. If we now rotate the ceiling away, we are left with a floor and a potential orthogonal transversal lines within which visible objects will arrange themselves once we add them in. So again, you can rotate the ceiling now away from you as well as if nothing was there. But as soon as you place something there, it will arrange itself according to these potential lines. Which are these blue lines here? So any objects again placed on the floor that have depth, i.e. they travel any given distance from near to far, will arrange themselves based on these potential orthogonal and transversal lines. If they do not have any depth, they will arrange themselves along the transversal lines. This is essentially what is seen with no ceiling, no walls or and no floor. So like it's like if you were in a giant room of white, as far as you can see, suspended on something very high. The only thing you have are unseen potential field lines. And if an object suddenly appears in front of you at a given distance or height, it would conform to these lines. So if you draw the orthogonal lines on an uncropped undistorted photo, you will start to determine the central vanishing point of the camera. Again, you can use diagonals or vertical lines and horizontal lines to kind of try to see where that vanishing point was for that camera system. So now let's add back our walls and some objects to see how they look. If we add a floor and a ceiling and we add in some objects, we get something that looks like this. If we add a right wall, an object would look like this along that wall. And if we add a left wall it looks something like this. So again the transversal lines are dictating the height and width of the uh rectangle. And the depth is according to the orthogonal lines. So note that the um if we have let's say that top uh rectangle or box that we have up there, note that one side is going along one orthogonal line, and the other side is going along the other orthogonal line. They're not parallel to one another. They are converging. All points are converging towards the vanishing point. So. And here's here we are with our, uh, ceiling rotated away and the objects are still there, but we've removed the walls and the ceiling, so, um, so going back here, you can see that we have our ceiling and our floor and our walls and our objects within our corridor. And now we're going to rotate the walls and the ceiling away and leave only the floor, but the objects themselves will still maintain their proportion. Another way of seeing this is that these lines dictate how the world is viewed. If a floor is placed below us, it conforms to these lines, and if a ceiling is there, it conforms to these lines. These lines are how we see the world. So in this photograph below, we are no longer have a tangible, physically obstructing wall or ceiling, yet our vision still contorts any non-uniform object along these lines. And although we have a vanishing line, you can see how the clouds are compressed along these lines to the vanishing point. Even the clouds are converging towards the orange point in the middle. But keep in mind that all of these are parallel structures to one another in in the linear world. And so, uh, you have to kind of spread these, you know, in your mind's eye, I guess you have to spread these things outward. Um. To to represent how they are in real life. Um, we'll go through this a little bit more as we go. But, um. In other words, if I just pick a point slightly to my left of the to the left of the orange point in the distance, and I try to walk towards that point, you'll you'll never reach that point along that straight path, because that point, that object just to the left of the orange vanishing point is actually, um, possibly many, many miles away to your left. But that point is converging to your right, just like all the objects on your right are converging towards your left. If that's not apparent now, hopefully it is by the end. But remember, we are converging all the objects within the world towards a central point. But they aren't actually there. They are spread out. So you can kind of see that here. When we place objects within our view, you can see how they move along these lines. Um, it's like looking at an x, y, z axis essentially. So objects with a lot of depth from near to far will appear to converge more toward the vanishing point according to orthogonal lines along a z axis. We'll call the orthogonal lines our z axis. Objects with only a little depth along your peripheral vision, or objects directly in front of you, align themselves at these transversal lines or the x y axis. And so I've kind of outlined some objects within this photograph. You can see some of them taking the orthogonal lines predominantly, which are the um, uh, the, the train station, uh, kind of lines in the railroad tracks. And those kinds of objects are moving along orthogonal lines. And then you have the railway, uh, ties themselves. You have some kind of rectangular, uh, box platform, uh, cement structure in the middle. Um, and that is going along the x, y, uh, coordinate or those transversal lines. So if I added in all the lines, you would see the linear world in which we exist, so represented in perspective. Uh, so again, all points converging towards a central point but proportional. So if I remove the depth from an object along my periphery, like I remove some of its z axis component, I guess you can see how it might look. It retains its x y dimension and the z dimension is lost, let's say. So it would if. Let's say we just removed more and more of the depth of the object. It essentially just looks like a rectangle again. Um. So note as objects move toward my vanishing point, their perceived depth is compressed more than their height or width. Ah. So again, if something starts out to your left like a car, it's going to have a certain amount of depth. But as that car gets further and further away, the amount of depth that you're going to see on that car is getting less and less and less, where essentially, at some point you're only going to be seeing the back of the car. Um, also, it will always appear to move to your right, even though it started out to your left. Um, because it's moving towards your vanishing point. Um, but in reality, it's always to your left, okay? It's always to your left, moving in a straight line away from you. But it will appear to be moving towards your right or towards your center of view. So we know some of the issues now translating the tangible physical world to our visual world. Let's now move the eye or aperture height of the observer, so that we are no longer perfectly positioned in the center of the corridor, and we see that our angles now change in our orthographic side view. So our lens is now moved lower to the ground. And you can see that the angles that are coming off the same exact objects or paint along the, the, the, the walls are now differentially compressed. So as our angles change, the information at the top is now expanded above our eyeline, and the information below our eye line is compressed. Um, if we add our paint back to our hallway, you can see the angle differences for the same ten feet worth of information ten feet below our eye line no longer represents the ten feet above. They are no longer equal to each other. They are differentially compressed, the angles are different. The size is then viewed differently, as we'll see. So our converging model looks a little bit different. Now things below the eye line compress at the same rate relative to one another, and the object sizes in their distances within that field, but not in proportion to the things on the other side of the eye line. Also, as we move lower and we lower our eye height, the angle at which the floor meets the ceiling increases because the ceiling is coming in at a steeper angle or steeper incline, so the ceiling meets the floor sooner. This is why when we lower our eye height, we can't see as far. Conversely, when we raise our eye height, we can see further the angles change. This concept will become more apparent as we go. So again, um. This if we reach a a, uh, vanishing point or a vanishing angle below our eye line. Um, and we move down and we are compressing those angles. We are at the same time expanding the angles above. These angles are paired, as we'll see. So, um, if we compress one angle of, of ten foot of paint below us and it's the same distance as the other ten feet above us, the one above us will be able to see more of it than the one below. Um. So the what ends up happening is that the, uh, vanishing point for the, uh, bottom amount of information is reached, uh, uh, sooner. And this makes the top portion, uh, essentially come down at a steeper rate, uh, to meet that point at the vanishing point. What ends up happening then is that we bring our horizon line or our horizon distance. Excuse me. Uh, in further, we cannot see as far. So information with this, this gap that's this orange rectangular gap here. So information within this gap represents what is lost when we lower our eyeline and our vanishing point arrives sooner from a greater flow angle. Compression. The amount of distance lost is proportional to the height loss in a linear relationship. So recall from earlier that the height was inversely proportional to the distance and h one over h two equals d two over d one. So and here's an example of of this relationship. If we're six feet up and a boat is traveling away from us and disappears at three miles, a rise in elevation of 12ft to 18ft up, so we're adding 12ft to our six foot elevation to now 18ft, um, would represent a 200% gain, and we would extend the horizon 200% to nine miles again, 1 to 1. The parts of the boat differentially compressed by the height change would be visible, visible again. But this isn't because you're seeing over a curve. It's because you can resolve more information with resolution angles that are no longer as compressed as they were previously. So again, you're decompressing these compressed angles to the point where now the information is is resolved and the light rays are now more resolved. And you can distinguish the object again. So continuing on with the differential rate of information loss, you can see the effect here. With these bottles the horizon is set by the observer height. So as an example, if you can see nine feet at a three inch elevation above the table and you lower your height 33% to two inches, their lower horizon is now closer by 33% and is now at six feet. And this represents the new lower horizon line. So again, vital links in the description. The rate of descent along the tops of the bottles is greater than the rate of ascent along the bottom, the compression rate is higher along the bottom, and the angles are more open along the top. Again, we have compressed angles along the bottom, which correspondingly increase or expand the angles that we're looking at. The tops with things will appear to descend above our eye line at a faster rate than things ascending below. So you can see the yellow line descending now at a faster rate than the ascending rate of the blue line below. Therefore, things below our eye line will appear somewhat stable in relation. In other words, it will appear like some a bit more of a dramatic decrease in in size or in height for the objects on top, whereas the bottom will appear somewhat stable even though the, um, the angles are more compressed, as we'll see if information continually moves into a narrower and narrower angle, eventually the lower part of the bottle will be unresolved while the top still remains. If I were to extend these bottles along a very long table, the bottom would disappear first with a lowered observer height, yet the top would remain. If I extend the table even further, the entire bottle will vanish when the top of the bottle descends below the lower horizon line. Again. If it's not apparent now, it will be as we move on. So the lower horizon moves closer in when we lower our height. So note that we cannot see the horizon gain or loss along the z axis with this perspective view. The vanishing point is the same point in both representations, but when you model these same angles as representative of the compression rate change orthographically from the side view, you see the horizon change along the z axis, so it will be the same as the height change, the amount of which the floor angle compresses proportional to the distance loss. That is essentially the bottom line. And this is a linear relationship, not an exponential one. So again, our perspective view here is on the left with our potential transversal and orthogonal lines in our, uh, horizontal eyeline and the descent and ascent rates, uh, represented in yellow and blue along the bottles. So with an orthographic view. Uh, looking over here on the right, if the heights compressed 30%, the horizon distance gets closer by 30%. So as an example, a bottle on the street was filmed 107ft away at different heights. When the camera is lowered, the bottom is cut off, the bottom compressed until the information vanished. With a higher viewing angle, the object comes back into view, into view in entirety and again. Video link in the description. As we'll see if the angular resolution of the observer is unknown, you can calculate it for the amount of the object that is cut off at a known distance. After a discussion of angular resolution limit, we will return to this example. So again we're showing that up higher the bottom will be seen. The angles along the bottom are no longer compressed. Once we compress those angles and we are viewing something, or let's say we lower our eye height to view something along compressed angles, um, the horizon along the bottom is brought in, uh, closer to our eye, and the bottom will vanish while the top will remain. So again we're showing the bottle here. This is the close up view. You can see a little bit of refraction distortion along the bottom of the bottle. But you can see that the bottom of the bottle is cut off. And the what you're seeing below that is what's called an inferior mirage. And we will go through that later. Um, but where it starts to mirage or it starts to mirage, that is the horizon line. Um, and you can see that that part is cut off. So as we'll see, the loss of information is a function of angular resolution and perspective and not curvature. So here's another example. If you're looking up an extremely tall building with a narrow viewing angle, by getting very close to the building and looking up a person beyond the vanishing line at the tallest floors could wave a flag out a window and not be seen. You can see that the side of the building becomes so compressed that the top floors become indistinct. If these buildings were a bit taller, or the person was standing a bit closer in the top floors, and anything sticking out of them aren't observed. So if someone were outside painting windows and ascended to the top floors beyond your eyeline or vanishing line, they would disappear building side first as they appear to set into the building. The building isn't curved. This is a function of perspective and angular compression of information. And again, these buildings are examples. They need to be a little bit taller or the photo taken a little bit closer in in order for us to lose the top floors. But that would be the effect the painter, uh, or the window washer would essentially set into the building as you got closer in. So here's another example of the observer height effect. And again video links are in the description. Uh, but when lowering or raising your iPhone, which has a small aperture in comparison to, say, your eye or a high end camera to view a flashlight on a very long floor like a supermarket. The flashlight, if it's low to the floor, disappears due to a limited viewing angle. So again, you can have your phone, uh, with the camera side down, uh, along the floor, uh, and very close to the floor. And you can do the same with a distant light source or any object for that matter. And you can raise and lower either your phone or the light source, and you will not be able to see it when both are on the ground at the same time. So raising the flashlight carries the same implications as lowering your eye height. The light is shining parallel to your, uh, position. Keep that in mind. That light is shining essentially in a direct line straight to the, uh, or what you would think would be a direct line, straight to the, uh, to the phone. But the phone camera cannot pick it up. It cannot resolve the light. Uh, it has hit its angular resolution limit. So on the left side picture, you can see that the light is about eight inches off the floor and it's visible on the right. The light is lowered and placed on the ground. Uh, and it's about a centimeter up and it disappears. In essence, it is set below the eye line of the observer, which is the camera within the diffraction limit, almost like a mini sun. So we'll continue going through this, but I hoped I explained the altered height within perspective when viewing a fixed object. And let's now see how moving objects within a fixed perspective are observed. So again, in the previous examples we have, we might be moving our observer height within a fixed corridor or some other landscape. And now we are going to. Uh, try to put objects within a fixed perspective and see how they travel off into the distance or get near. So we return to our center line view will keep all things equal, and look straight on at an object. If it's directly in front of me and I just move backward or move the object further away, it maintains its proportions from left to right, top to bottom, and just gets smaller. So just like before, it's just going to move straight away along the eye line, straight straight towards the vanishing point and diminishing in size proportionally in all dimensions. It's just like if I put transversal lines like which are our rectangles, and I pulled them out and they are taken out and just modeled as if they are moving rectangles. So you can kind of see the effect here. It's getting closer. They're getting closer. They're getting closer. They're all in proportion to one another. They're getting further. Getting further. Getting further. All in proportion. Um, if we have an object now to our left that is of sufficient size to have aspects above our eyeline. Um, you can see that the top moves toward inward, toward the top orthogonal lines and the bottom along the bottom ones. And the compression expansion rate is the same between the top and the bottom half. So we're going to keep this all to the left of our eye line or excuse me, all to the left of the vertical orthogonal line. And you will see that it will always remain there. It will always maintain itself to the left of that side. If it starts out, uh, below the eyeline, it always is below the eyeline. If it starts out above, it's always above. So on and so forth. Um, and again, you just model these according to orthogonal lines, uh, descending and ascending, uh, or converging left, converging right, whichever dimension you're looking at it from. Just, just take the rectangles and, um, proportion them or stratify them along these, uh, orthogonal lines to get your effect. But you will see that they never cross the point at which they, uh, uh, vanish. Is, is going to be that center point. They never cross over to another side or another quadrant of the image. So let's see what happens with moving objects that are not directly in front of you. Um, or to your left, let's say, let's say they're now below your eye line and to your left. So both they're, um, in this case, we'll assume our moving object starts out to our left and below. If I maintain my eye height and the object moves away, it will appear to converge toward the vanishing point. So our object on the floor appears to rise and move to the right. But again, it is moving straight away from us along the floor. Note that because the object started out completely below our eye line and to our left, it will always remain below our eye line and to our left. It will just get more and more towards the center point. It will diminish according to those field lines set by our observer position relative to the floor, walls and ceiling. In other words, its ascent rate, descent rate, left and right converging rate. All those rates essentially are set by those angles of the orthogonal lines. If I lower my eyeline, you will see how the orthogonal lines, uh, start to change. And the rate becomes differential. And you also see that the transversal lines also get more compressed or less compressed, depending on, uh, raising your height or lowering your height. So before moving on, let's let's keep in mind that with our eyeline move lower, the building still maintains its proportions relative to itself. So this is an important part before we move on, if we pull out a rectangle like the green or the yellow ones, let's say so instead of the orange and blue paint, we now just pull out one, uh, yellow rectangle we'll say, and one orange or excuse me, one green rectangle, uh, far away. Um, there's still the same rectangles, but they their position has changed. And as you see, the eyeline is no longer centrally situated within the object. It's now lower. So as we move our, um, our observer height down, the vanishing point, uh, moves down relative to the object, but the object itself maintains its proportions. So we can look at our bottle example the same way the bottle is still the same bottle. Uh, it's just viewed at a different, uh, compression rate. Um, so you can see how the eyeline is in the middle of the bottle on the image on the left, but the eyeline is moved down, uh, towards the lower half of the bottle with the image on the right, but the bottle is still the same bottle height. Also note here if you go to the right side perspective view, you can see how those lower transversal lines are very close to each other, how the upper transversal lines are much further apart, whereas on the left perspective view they were all equally proportional to one another. Again, this is a function of your observer height. It's going to move your horizon line in or out, depending on the expansion and compression rate of those angles. So with the lowered eye height, um, we can see that if an object starts out below our eye line, it still remains there throughout and maintains its relative proportion to itself. Here's our box moving away from us, and we've got it very close to the eye line starting out. But you'll see it never goes above the eye line. It is always below the eye line. It just gets smaller and smaller. Hopefully that makes sense. I'll look at if you also know it's stratified along these potential orthogonal lines. Uh, so. All right. Now we'll extend the object's height so that it starts out with some aspects of it above our eyeline, but most of it below. So now this is now this object is a little bit bigger. It's in front of us. It's now partially above our eyeline and partially below when it starts out. As the object gets further and further away, the aspects above the eyeline will compress at a different rate to the aspects below. It will appear to be a more dramatic change at the bottom, because more of the bottom of the object was below the eyeline to begin with. So again, just take out some rectangles, stratify them along, uh, the orthogonal lines for their rate of ascent and descent, uh, along some of the orthogonal lines, like I said. And you will see the effect. So you can see here in the bottom left, those are the rectangles that I pulled out, uh, from the uh, perspective view in a central perspective view. And then you just move the corners so that they stratify themselves along the, uh, orthogonal lines, which I outlined here in black. And you can animate them if you want, and you can push them off into the distance or make them come out to you as the observer. Either way, they will maintain their relative proportion above and below the eye line throughout the entire way. Let's reverse things now with most of the object above our eyeline. Like if we were about six feet off the water and looking at a large boat traveling straight away. Um, and again, it starts off a little bit to our left. Uh, if you look closely at the diagram, you'll see that the bottom of the object is approaching the vanishing line or horizon. And at a certain point, as the object near this line, the angle is simply too compressed. You'll see. You'll see down there at the bottom that the blue outlines of our rectangles start to merge along the bottom, and we go from distinct lines to a single kind of blurred merged line. Meanwhile, the top of the object still has distinguishable lines. We can count these rectangles much more easily. The bottom information is being lost as an object moves straight away from us along a plane. So again, I can blow up this image here and you can see it there on the lower right. The image integrity is somewhat lost at the bottom, while the lines are still distinct above the eye line. I can see, uh, the distinguishing lines above the eye line of that object. I can no longer distinguish them below. Um, as we'll see, the compressed angles below the eye line have proportionately expanded angles paired above. So as one compresses, the other expands. Keep that in mind. Or as I move left, uh, and compress the left angles, the angles on the right expand, so on and so forth. Uh, also, for an exceedingly tall object, the aspects below the eye line are essentially negligible in comparison to their total height. So they stratify along the horizon line, like if I was looking at a, uh, mountain exceedingly tall, way higher than six feet of the observer, let's say you can essentially view it as being all above the eye line all along the uh, uh, horizon line. So, uh, again, also objects. They descend at a rate commensurate with their pairing below the eye line. So, uh, again, if I compress the lower angles, that eye line or that orthogonal line is going to become more shallow, but at the same time, the orthogonal line that is paired above will descend at a higher rate. So again, if close to the ground, the descent rate is steep. If higher up, it's more shallow or gradual. So in the orthogonal view here on the left, these mountains would represent the same height if I observe them that way in that side on orthographic representation. But they are seen below, uh, here in a one point perspective. And so they therefore must diminish in size according to the transversal lines. I could replace a 4000 foot mountain with a four foot car parked at different distances, and it is the same phenomenon. I would just be lower to the ground relative to the car to view it this way, and put my horizon line along the lower edge of the wheel, and it would be the same exact thing. The rate at which they diminish is linear within a fixed perspective. In other words, double the distance half the size, triple the distance, one third the size, and so forth. We can reverse some of the near and far objects, keeping them according to their transversal lines, and we start to see a typical panorama form with six equally sized mountains. The three on the left are in a straight line, but they are converging right. The three on the right are not in a straight line, but have the same heights at the same distances as the ones on the left. We could add in some clouds, different and some different mountain heights here and there, and you can stratify them according to these orthogonal and transversal lines in the same ways. And this is how we see the world. So to summarize the lens, the lens converges light rays that expand from a source according to the inverse square law by a proportional amount. So the light expands out toward us exponentially and is then converged behind our retina, uh, to a focal point, um, and inverted along the back of our retina and sent to our brains to reconstruct the image in a linear fashion. Uh, the rays with which we view an object have different angles depending on the presence of an obstruction, like the ground, ceiling or walls. So we compress them depending on our distance away from a tangible object. In other words, if we are close to the floor, uh, we are viewing an object through more and more compressed angles. If we are close to a wall, the same thing happens. And likewise, if we are close to the floor, the angles that are paired to the ceiling above are now more open and more expanded than the compressed angles below. Same thing with the walls. So again, the information above and below or on either side of the eye line are paired and as one angle opens, the other compresses with a lowered eye height, the ascent rate right. The rate of the blue line in this example, uh, decreases below our eye line, and it becomes more stabilized and the descent rate increases above our eye line by a corresponding amount, and things appear to descend at a faster rate. We quantify this phenomenon below, the amount by which we lower our observer height is directly proportional to the amount by which we bring the lower horizon line in closer, and vice versa. If the lower horizon is closer with the lowered height. Once that distance is met by the descending information above our eye line, things above will set into the lower eye line with further distance, and we were able to quantify that as the h one over h two uh, is equal to the d two over d one uh distances. So if I take my height and lower it by a certain amount from h one down to h two, the horizon distance will come in uh, by a directly proportional amount. Keep in mind within these within this diagram, it's conceptual. I am not implying that the horizon line within the lower picture has moved to the left. Or excuse me, the vertical vanishing line has not moved to the left. It is still within the center of the picture, just like the vanishing point is along the top picture. Um, but this is just conceptual to show you that they move in by that amount. That amount that you lowered is the same amount by which you decreased the distance. So with that understanding, let's now take a deeper look at the angular resolution limit. Recall that our we have a lens system. And we will now complete that picture and add the bottom of the tree. So before we were just looking at two rays of light, uh, coming off the top of the tree, we're now going to put the bottom of the tree on and we'll look at the two rays coming off the bottom. And that completes our picture of our inverted image behind the lens, which is then projected onto the retina, the cones and rods on our retina, uh, especially near our fovea, which is the central point, um, where light is most, uh, uh, easily resolved, the highest concentration of our rod and cone cells. And then it's transmitted through our optic nerve back to the visual cortex. So, um, as the image continues to get further away. So now looking at the middle diagram, the angle, the angle of resolution across F1 gets smaller and smaller. Uh, as you can see between the two pictures, as we've moved this tree out to our left, that angle is getting smaller and smaller until it gets to about 0.02 to 0.03 degrees, or one arc minute of the human eye. And this is the point that the cones in our retina can no longer resolve the image, because the light rays are too close. In other words, we have met our vanishing point and the information cannot be resolved. And this is again the visual acuity of the average person taken from the central portion of the back of the retina, where our rods and cones are most concentrated in our fovea. Light hitting the fovea at less than 0.02 to 0.03 degrees cannot be distinguished from other points of light, and it is too compressed. More specifically, it is the point that two light rays diffract to a point of 50% overlap, which we will get into. So six rays become three, four rays become two and two become one, so forth. And an image cannot form with one ray. It needs the other to create the inversion at the convergence point. So no image forms at that point. So with this broader concept in mind, let's look at the diffraction limit. So the formula for angular resolution limit is given by theta in radians equal to 1.22 times the wavelength, uh, divided by the diameter of the aperture. The angle is proportional to the wavelength of light divided by the diameter of the aperture, like it says in the formula multiplied by a constant 1.22, where that constant comes from. The constant is derived from a mathematical construct called a first order Bessel function, which is used in a variety of areas, not just angular resolution, and it is conceptualized similar to ripples from water spreading out in a circular diffraction pattern, just like shown in the picture there. Uh, has waived minimums and maximums that decrease in amplitude. They spread out into the distance and eventually drop to zero. Uh, it's somewhat like a sine function, but with progressively decreasing amplitude. Uh, almost like a damping effect. So in this case, instead of water, the diffraction pattern is made by light scattering through an aperture or a pinhole and the, uh, circle circular ripple pattern is called an airy pattern. So the airy pattern is there on the bottom right. You can see how it looks very similar to the diffraction pattern of a drop in a pond, let's say. So just like with water, the wave amplitudes, they have wave amplitude minimums and maximums, and each of them has an increasing diameter as it moves outward. The central maximum, which is the bright circle or sometimes fringe with light in the middle, is called the airy disk, and the dark circles are the minimums. So the bright circles are the bright fringes. The dark circles are the dark fringes. The very bright central disk in the middle is called the airy disk, and it's the most concentrated, uh, maximum of the diffracted light. So the radius of an airy disk can be derived from R, the radius equal to the length of the light after the aperture, diffracts it and before it hits the detector. Uh, times the sine of theta or that angle, um, where L is again, is that length between the aperture and the image as depicted. So if the aperture diminishes, there is more diffraction and the airy disk enlarges. That's just how light works. So if you reduce the size of the aperture, the airy disk gets smaller. Um, almost like it's spewing the light out, uh, you know, kind of splattering it all over the place, I guess. And as it gets bigger, the light can create more of a of a pinpoint, uh, airy disk pattern on the back. So larger the aperture, uh, smaller the airy disk, and smaller the aperture, larger the airy disk, smaller the aperture, uh, more diffraction and larger the aperture, less diffraction. Okay. So from Young's double slit experiments and Huygens principle, we know that light traveling through an aperture and diffracts as if it were many apertures, and it essentially interferes with itself, yielding the same diffraction pattern as multiple slits. And I'm not going to get into the specifics here, but by dropping normal lines from converging rays, traveling different distances to construct the diffraction pattern, a formula for the diffracting angle theta can be derived such that theta in radians equals 1.22 times the wavelength divided by the diameter of the aperture. If you want to look up the derivation of this formula, uh, there are many videos online describing it. So this angle theta, which represents the radius of an airy disk, would also represent the distance at which two distinct airy disks would overlap by 50%. So when two non-uniform light sources are used and we have two distinct airy patterns, we can resolve the two disks until they are overlapping by 50%. So this is called rayleigh's criteria. Once the airy disk overlapped by more than one half, the image cannot be resolved. And we'll see that here in a minute. The two objects become one, and one is missing. If the disks are larger with smaller apertures, they will overlap more quickly and will be lost more quickly than the smaller disks with larger apertures. So therefore the smaller angle, or the smaller theta, the better the resolution. You can pinpoint them as two distinct units for longer. So within our variables, larger wavelengths and smaller apertures diffract more and are less resolved versus larger apertures and smaller wavelengths that diffract less and are more resolved. So here's the depiction. Here. You have 0% overlap with these two airy disks. We'll say. Then there's 25% overlap. You can still somewhat resolve them. And then at 50% overlap they become non distinct entities. And they basically merge and blur into one pattern. So resolve somewhat resolved to basically unresolved. And this distance where it's somewhat resolved um uh just before 50% essentially is the Rayleigh criteria. And this distance happens to be the same distance as the radius of the airy disk. Right. So 50% overlap of two disks is the point of the radius of a single airy disk. So as the patterns overlap, you'll notice intersections between the bright and dark fringes. So this kind of criss cross pattern that the minimums overlap with and the maximums overlap with creates interference patterns, um, between the bright and dark fringes. And these are constructive and destructive interference zones. So as the two patterns get closer the interference points will start to lessen in number. In other words, you'll have a whole bunch of these criss crossings as they start to merge with one another. But as they keep merging, they'll be less and less of them to the point where we're down to essentially one, uh, big overlapping, um, uh, constructive and destructive zone. So essentially the patterns will start to look more and more similar as these two points merge. And this will become apparent when we give a little example. So here is our two patterns. We'll say of our airy disks. You can um, I've outlined in black here, uh, put lines through the constructive, uh, zones where the lines cross one another. So you can see there's many patterns when the two airy disks, which are these central disks of these two patterns, um, are far apart. There's a lot of constructive zones, just like up here in the upper left where I just overlap the two disks. Um. The. As we'll see, the constructive zones will start to spread out, uh, and lessen in number as the disks get closer to each other. Also note that the I've depicted here, uh, the blue lines that are these concentric circles are, um, all essentially equal in magnitude. But keep in mind that with an airy disk, these, uh, actually diminish in, uh, relative strength as they move outward, but the same effect occurs. So here we are with the discs a little bit closer. You can see between these two we're getting less constructive zones if I move back and forth. And then let's move to the next one. Now we're getting less constructive zones. They're spreading out more and more as well. So let's just continue on and move through some of the images. And here we're down to three constructive rays. And the pattern is spreading out uh down to excuse me. On the last one we were down to five. Now we're down to three. Uh, you can see in the upper left we're down to three. And here we're just prior to 50% overlap. And already it's difficult to distinguish the pattern as two distinct patterns. Now we're at 50% overlap in the observer perceives one pattern. Also note that we also only have one interference zone extending vertically at this point. You can also note in the upper left it looks as though it's one pattern. So if we return to our lens diagram, you can see how two rays merge with distance and height above and below the eye line. So as the object gets further or shorter, the two rays extending through f one prime and f one get closer and the information overlaps. So again with further distance or with the object shrinking in height, the object behind our eye line, the projection behind it, those two rays get closer and closer together, so the stump of the tree or the bottom of the tree below our eye line has a less height than above the eye line, and so you can see the two rays are a little more closer together behind the focal point, f one. And I've depicted those merging rays with our airy disk pattern merging, um, that we just went through before. And you can see how the constructive zones get, uh, less and less constructive zones with uh, uh, more and more, uh, angle divergence until you get to the unresolved 50% overlap point. So again, at 50% overlap, any aspect of an image will be lost if it is within theta and images converged behind your eye with at least two rays above the eye line and two rays below. So if the two convergent rays below the eye line overlap by 50%, the two rays become one. No image convergence occurs and the bottom of the object disappears. And then just some notes here before moving on. The two light sources send out any number of wavelengths along the visible spectrum. Keep that in mind. Once they hit an aperture, they separate. And so we are examining two coherent sources interfering with one another the Reds interfering with the reds, the blues, with the blues, etc.. Below is the diffraction pattern of white light with a red laser diffraction overlay. The red is lining up along the red of the white light diffraction, and so on. So with a cursory understanding of airy patterns, using our formula and some variables, we can calculate the smallest resolution angle theta for the human eye. We can then take that angle theta and apply it to any distance d to calculate what the smallest diameter or height of an object is that we can resolve. So let's do a practical example indoors before moving outside to a large distance. We'll see if we can see a bug. Uh, small bug 20ft away for example. So we'll first get our, uh, theta angle. Uh, the at which it's at meets rayleigh's criteria of 50% overlap. And so 1.22 times the wavelength divided by the diameter aperture gives us our theta. And then once we have theta, uh, we can use it to. Calculate what the height of the object is at a distance d that we can resolve. Keep in mind that this angle that I've represented here is a right triangle within our eye, uh, to the object. But in reality it would be more like an isosceles triangle. But because. We're using such small angles, uh, in comparison to the, um, to the object. The two distant angles, uh, are essentially close to 90 degrees. And so we'll use the right triangle. So tangent theta is equal to height divided by distance, and height is then equal to the distance times tangent theta. So we'll say our bug is two millimeters. The wavelength of light we're observing here is 550 nanometers, right in the middle of the spectrum. Visible spectrum. And our pupil aperture diameter is four millimeters. So our bug is 20ft away or 6.1m. So we first solve for theta. We plug in our numbers. Make sure you convert radians to degrees which is 57.3 degrees per radian. And you come up with a number of 0.00874 degrees. So. If we then use that theta to plug into our, uh, triangle. As the angle. We can then solve for the height uh, across any distance d and in this case the distance d was uh, 20ft or 6.1m. Again, make sure that your units are the same. And so, um, uh, plugging in for height 6.1. Uh. Uh meters distance, uh, times. The tangent of 0.00874 gives us a number of 0.9mm. So if our bug is two millimeters, we can see it. If our bug is less than 0.9mm, we can't. So the average angular resolution limit of the human eye is 0.02 to 0.03 degrees, or one arc minute. So 0.00874 degrees is definitely the best case scenario. And keep that in mind when there are no aberrations in a lens or other distortions altering the system, we are only limited by diffraction or diffraction limited uh, since aberrations abound, though for our purposes we will assume the standard one arc minute for the human eye. So let's go out to a large distance. Now the diameter of the pupil ranges from 2 to 4mm in bright light and 4 to 8mm in the dark. The wavelength of light ranges from 400 nanometers of purple to 700 nanometers in the red. Uh, the orange and red wavelengths are less resolved than the blue. Keep that in mind. But they are also, uh, less subject to refraction and atmospheric scatter. So those higher wavelengths have more diffraction, but they are also less refracted. And we'll get into that later. So with theta at one arc minute, we can now see how this resolution angle across the distance D limits the height or diameter of an observable object. We can do this over any number of distances and solve for the smallest height as d tan theta. On average, we lose four feet at two miles, six feet at three miles, eight feet at four miles, ten feet at five miles, and so on. And this assumes one is either indoors or outdoors with compressed lower visual field along the ground. In other words, placing an object within compressed angles such that theta is reached for that object. So holding theta constant, we can also reverse this triangle to find out what the outdoor horizon distance is for a given observer. So I just reverse the triangle around and gave it. Now the object height is our eye height, and we want to solve for the distance d at which we can resolve something. So recall our horizon calculation. Uh, we have our diagram here where h one over h two. The ratio is equal to d two over d one. And those are held within the same angle. They are roughly approximating the same theta angle. So if I am at height one with the green person and I extend myself higher up to height two, uh, I can now go from distance one to distance two, uh, holding theta constant between the two. And we can do this over any number of heights and solve for the horizon distance equal to the height divided by tangent of theta. So plugging in, we see that, uh, for uh the height we lose four feet at two miles, we lose six feet at three miles, we lose eight feet at four miles, and we lose ten feet at five miles and so on. And if we want to see how far we see at given heights. So for distance, we see two miles at four feet we see three miles at six feet we see four miles at eight feet, five miles at ten feet, and so on. So if theta is only met below the horizon and not above, the object disappears below our horizon and not above. Keep that in mind. Recall that the angular compression rate is determined by observer height. So if the compression is met next to a wall, then then object along that wall will disappear. As the object gets further away. So here we have our equal floor and ceiling compression rate. So again, instead of being close to a wall, we're right in the middle of a corridor. Again above and below the ceiling is the same distance that the floor is away from our eye, and the walls are the same. You could say two and we would equally reach theta for a given object above and below the eye line. So the loss of information occurs only for information within that 0.02 to 0.03 angle, if it's reached at that wavelength and at that pupil diameter. So our horizon represents all information contained within an angle less than 0.02 degrees above or below the eye line. Or put another way, our horizon is comprised of plus 0.02 degrees to -0.02 degrees on either side of the eye line. And again, if we did differential floor and ceiling compression, and we got closer to a tangible physical wall or a physical ceiling or the physical floor, then we would have differential angle compression and anything met along that compressed angle side, either the wall or the floor or the ceiling is going to reach theta and it will be lost against that object. So if we reach theta for only the bottom of our eye line, a six foot person is completely lost at three miles and at 18ft, and an 18 foot tower is only partially lost. It loses six feet essentially, and it loses it bottom first if we are compressed against the floor. Also recall our building example. There is a concept of a personal horizon and what we believe to be a true or physical horizon. But as far as we can tell, the sky never actually meets physically meets the floor. But when we match up our eyeline to a point roughly parallel to where the ceiling and the floor appear to meet, we can see these optical phenomenon better and see the, uh, you know, object disappearing bottom first. Essentially, we're provide a better backdrop to these phenomenon. But keep in mind angular resolution limit exists everywhere, no matter if we're looking at the horizon or straight down at our feet. Just like if I am looking at a human hair on the ground, there is a point where it will not be seen and that point is theta. If I look up the side of a tall building, the building becomes my floor and the same effect occurs. So if the building were, say, greater than 5280ft or one mile up where you are close by and looking up, you won't see a person waving out the top windows at one mile. Uh, recall that we lost two feet at that point, so keep that in mind. And again, keep also conceptually, these buildings become the visual floor and ceiling of our vision in this depiction. So an object within theta near our horizon would be lost according to the same rules. And keep in mind these buildings are also parallel to one another. They are not curved. Yet if the buildings were taller, you would see a window washer disappear building side first as he ascended. So with that in mind, let's see what happens with a lowered eyeline and an object just to our left. Returning to the principles of perspective, you'll notice the more dramatic changes in height are seen at the top than at the bottom of the object. When we lower our eye height, just like with the bottle example. So as the object gets further away, the changes we observe at the bottom somewhat stabilize relative to the top, and we see things descend top down more dramatically than we see things ascend bottom up. So again, just pull out some rectangles and place them along these orthogonal lines and stratify them to the vanishing point and you will see the same effect. So if I zoom in on these buildings that I or these multicolored buildings will say, um, and pull out what is zoomed in on, you can see that the lower transversal line, these transversal rectangles, essentially this lower aspect of it that was compressed differentially to the top, it approaches the purple horizon line faster than the top does. So theta is encountered just prior to the eye line, and zero information is transmitted on the next image from below. So essentially it got cut off. You'll then only see the top continue to descend. Um. Top down. So, uh, the yellow, green and orange and gray blocks that were below the purple horizon line to start out with, all those essentially disappeared. Again, this is not the curvature phenomenon of physical curvature phenomenon. This is how your eyes work. So one note. Also, this is difficult to simulate perfectly because the software wants to always keep at least one pixel when you start to shrink things down, and it is best to physically separate what is below the eye line and what is above, then compress them along the orthogonal lines from there. That is usually a better way to go. You'll notice though, when you do that, that you reach one pixel along the lower orthogonal line a lot quicker than along the top. And just to reiterate the point, an object never completely disappears until two theta is met or the top sets into the bottom horizon, um, or an opposing horizon line. If we're viewing something along wall, let's say. But the two common methods would be if two theta is met or the top sets into, um, the lower horizon, since the lower horizon is usually much closer in than the aspects above our eyeline. In other words, the floor is usually compressing an object much more than the ceiling, since the ceiling in the case of the sky is incredibly, uh, or is much higher. So and even in that case, the case of also of two theta being met simultaneously is also very rare, because for that you need the floor and the ceiling to be the same exact height away from your eye. Um, in order for that to occur, if you're slightly above, uh, uh, you know, closer to the floor than you are to the ceiling, then theta is met along the lower aspect of the eyeline, closer or sooner than it is above. Likewise, if you are closer to the ceiling than you are to the floor, theta is going to be met along the ceiling a lot quicker, and the object will disappear along the top. So, uh, remember there you need four raise necessary for an object that exists above and below the eye line. Uh, you need two on either side that converge the image behind the focal point on our retina. So within those four rays, you need two theta for, uh, simultaneously to to vanish an object, or you need one to be met and then the other aspect of that object to then set into that, uh, horizon line. Again. So here with our tree diagram. Here I'm representing the top and the bottom of the tree disappearing at two theta. But again that's if the compression rate is the same which is of course not the case outdoors. You are closer to the ground than you are to the ceiling or sky. So we may perceive that we see the tops of something far longer than the bottom, because the angles above our eye line are much less compressed than below. We still may have the same number of angles above and below, but with a lower observer height. The bottom angles are far more compressed, and the paired angles above correspondingly expand again. As one compresses, the other expands, but the expansion is still limited by the ceiling, so it still gets smaller. It just doesn't keep expanding. In other words, objects moving off into the distance will disappear from the bottom once theta is reached, but the top will remain visible, just like we've talked about, because its angle is bigger. As it then moves off into the distance, it will continue to set into that lower horizon line, either until its theta angle is met or it just completely disappears one or the other. The total number of potential angles above and below the eyeline are the same. Since we have the similar number of rods and cones circumferentially positioned around the fovea. But when we alter our observer position, the information alters. If we lower ourselves to the floor, the angles above the eyeline become wider and the angles below become smaller. As you can see, when you just, uh, fix some points, uh, along some lines and move a central fixed point along those two. You'll notice that the angles change size as you move the, uh, in this case, I, uh, vertically, you'll see that the angles above get wider and the angles below get smaller. Again, the total number of angles are the same above and below the eyeline. We only have so many rods and cones, but their compression rate, as we get higher or lower against the tangible object, starts to differ. Again, you can see here that the orange rectangle along the bottom diagram represents the horizon loss because we moved theta in or we met theta uh, sooner. For the lower compression rate, we moved our eyeline down, theta was met and the horizon moved inward. So let's develop this kind of paired idea. Paired angle idea further. Assume we have two equal balls at a given size nearby at our angle one. There we have about 12 angles along the top diagram. So we'll say at angle one we have this, uh, these two objects of the same size as the objects move away on a straight line. They are observed with progressively smaller angles. So again we're looking at something along the ceiling and something along the floor. It starts out at a certain size and as it moves away it gets smaller. Once theta is met, they both disappear. And that's represented by we'll say angle 11 here at the top diagram. So theta is met at angle 11. Just an arbitrary point we'll say. So anything beyond that point isn't resolved. You cannot see that object any further. If you were to collapse this diagram inward like we did our other ones along the central line, you would see that the vanishing point is met along that point, or essentially when we collapse theta to a point of being perpendicular to our eye, uh, nothing is seen beyond that point. We'll see that here in a minute. But let's let's alter our diagram now. And now assume the same scenario where we have two equal balls or objects at a given size nearby at our angle one, but the ceiling is very high relative to the floor. Now we've pushed the ceiling up much higher. The balls are still observed at angle one, however, the same ball above us is now further away, so it's smaller. The one below us is closer, so it's a little bit bigger. Theta is met now at angle eight let's say along the bottom because we compress those lower visual angles. But along the top theta is off the page and angle eight is much larger. So our angle eight. Now along the top you can see is much bigger in comparison to angle eight along the bottom. And if we try to find what theta would be, what angle that would correspond to, what number, I guess along the top it's off the page. It would be hundreds in the hundreds possibly. So the top theta is now paired with an angle beyond angle eight below it. And angle eight is what set our lower horizon distance already. And so it's never reached along the top. And essentially what you'll observe is that that top figure or object or ball in this case will now set into the lower horizon line. So again, in this orange, uh, rectangle we have here, since theta is met at angle eight below, this information is now lost to the observer. Anything within that, uh, orange rectangle and the horizon distance essentially decreases at that point. But again, you will you will still observe, uh, the ball, uh, set into the, uh, lower horizon. But we'll see this here in a diagram. So when observing this scenario, theta is met for my lowered observer height and sets my lower horizon limit or my lower horizon distance, which is now closer in at that angle eight. So for the information above my eye line, the descent rate is now steeper, just like with the bottles, but I have less of an angular compression rate, right? The angles are more open. So these angle pairs, the eights or the ones or the elevens, etc. must always remain in proportion, but diminished together according to the ceiling or floor distance. If one angle decreases, the other increases, but they are both diminishing with distance. Theta is never met along the top again because the horizon distance is set by the bottom. It occupies 50% of my vision. So wherever the information above my eye line is at that point, when these two halves collapse is what I see above my eye line. In my perspective, the top continues on independently, but you will never see the objects diminish in size and reach theta along the top, because they will have already set into your lower horizon. They are gone to your vision. Note also the steeper descent rate at the top, the higher the ceiling, the steeper the descent rate. Within our perspective, this is how our brain interprets it. The nearest ceiling, ceiling or floor sets our horizon. Similarly, the nearest wall would set the vertical vanishing line if we were up against a long wall, let's say. So if we collapse our two or our diagram here, and we press the image downward, um, and along the top and upward along the bottom to reach a essentially horizontal line. So you see that along the bottom, uh, the, um, angle eight doesn't have to travel very far to get to the purple line, but the angle eight above has to travel, uh, quite a bit more, uh, to do that. But essentially what you're left with is, uh, uh, basically what we see in perspective, as I've depicted here in the upper right, you have the larger angles, uh, resolution angles along the top and a steeper descent rate characterized by the orthogonal lines. And along the bottom you have more compressed, uh, angles and those closer transversal lines and, uh, uh, more stabilized or shallower orthogonal lines. You also have the object larger, uh, to start out with, but then gets rapidly smaller, uh, with distance because of the, uh, how close in proximity those transversal lines are. And along the top you have the object start out smaller, but it remains relatively stable because the angular size compression rate is not as great. So the angles remain somewhat large as it descends away from you, and it maintains its relative size. Uh, for the most part. And at the end, uh, it will eventually set into the horizon. So if you probably have guessed, the sun is a practical example of this. The sun and moon set because they travel a short distance in a straight line along the very high upper boundary. If we extend our ceiling to a tremendous height, you see that the rate of differential angular compression between the floor and the apparent ceiling is large, and the descent rate steepens dramatically. So the object descends to the vanishing line rather quickly while maintaining a relatively constant resolution angle as it travels. It therefore does not experience an appreciable change in size. Once it hits the lower horizon, it sets. Again, this occurs without the object having to travel very far because it is so high up and without an appreciable change in size. Stated another way, the higher something is, the less it changes in appearance. The distance it has to travel before it sets is dictated by how far we are seeing below the eyeline, which is in most cases about three miles. This is for a diffraction limited system, but remember, there are aberrations and refraction effects within the atmosphere that can alter the size of objects and distort them. We will cover these later, but minus these effects we do see a relatively constant sun and moon. To visualize this, let's return to our perspective diagram that we subtracted the walls away from and raised the ceiling to a tremendous height. We would see the lower angles compress and the upper angles expand correspondingly. An object traveling high up would now have very little change in angular size before it sets into our lower horizon. Again, we have met theta along the bottom, but we did not reach it along the top. But the object along the top has correspondingly expanded its angular size. If it is viewed high up in altitude and as it goes down, it will eventually meet the lower horizon line and set within it. So let's return full circle. Full, full circle to our light on the floor example. And we'll see that we have essentially created a mini sun. In this example, we are assuming the floor is flat and level. The person far away is simply placing the light toward the camera and raising and lowering it from the floor to their waist. The observer height, which is the iPhone camera with a small aperture and a larger theta, is on the floor, lens side down. The video is a bit fuzzy because it is seen zoomed in, but I took a screenshot of the native video and you can see that roughly 50% of the image is the floor below the aperture, and everything else above the aperture is the other 50%. It's just slightly tilted in that example, but you can see how the bottom is the floor and the top is everything else. The floor has set my lower horizon and some distant part of the floor has vanished. That is now my horizon. And the LED light that you can see I'm holding in my hand will set into it. We will see later that magnification does not alter perspective, and you cannot bring back information that is smaller than the angular resolution limit of the observer. Apologies for the movement artifact. It was my fingers trying to stabilize the tilt. It's a lot more stable when it's not zoomed in, but when it's zoomed in it's a little bit, uh uh, unstable. But, uh, note the, um, person here, and she is, uh, holding the, uh, flashlight pointing towards us, just raising it and lowering it, and you'll see it. It's on a loop. Uh, so it just goes back and forth, but you'll see it disappear and reappear, uh, to the observer. So I'm playing it here. You can see the light. She's putting it down. It's set, it's gone now you can see it again. Just pushing it down. It's set and gone. She's pushing it again. Um, so this said, the sun doesn't set entirely due to perspective. Uh, however, and there is likely a personal atmospheric lensing effect that explains star trails and other phenomenon such as reflections under clouds and so forth. But we will not cover that here. Uh, you can see video link in the description about those phenomenon. Um, let's now return to our bottle example. We asserted that if the angular resolution limit of the observer is unknown, you can calculate it for the amount of the object that is cut off at a known distance. Once calculated, you can use that angle to calculate a height change with a known distance change, or a distance change with a known height change. So we'll use Mrad designations, which are 1/1000 of a rad or radian, where two pi radian is equal to 360 degrees, or radian is equal to 180 degrees divided by pi, which is equal to 57.3. Like we mentioned before, this method is especially useful for angular size comparisons with the known distance or height within the same theta. You can use the ratio of the height to distance of the unknown object if the angle is the same between the two. In other words, if one object is. Within the hour, obscuring a distant object. They are assumed to have the same angular, uh, size, and you can then use the ratios to calculate the distance or height between the two. Uh, that's not what we're going to do here. But you can see the effect of doing some calculations on, uh, pictures. So we know the following within this picture, let's say we know the bottle is 3.5in tall and the observer is 107ft away. Uh, so, uh, it appears that the bottle has a height of about 12.5 milliradians or 3.5in per 12.5 milliradians, which comes out to a 0.28in per, uh, mrad. So, uh, again, these designations, uh, were somewhat arbitrary. I just took, uh, the amount that appeared to be cut off as one unit, and I stratified them up from there. Uh, and you can do the math from that point, approximately one mrad is cut off from the bottom or 0.28in. So height is equal to 0.28in and distance is equal to 107ft. Um, keep in mind that 0.28in, uh comes out to 0.0233ft. So we can plug that in keeping like units the same. And you get a number or an angle of 0.012 degrees. And this angular resolution is pretty good but represents this camera at the predominant wavelength and aperture diameter with which it was observed. So with a theta known for this particular object, you could predict how far you are with how much of the bottle is cut off, or alternatively, how much of the bottom is cut off by how far away you are. So we'll do another example after discussing some refraction effects to kind of make this a little more clear, but we also need to briefly discuss magnification I think, before we go further. So magnification can be thought of as the ability to see an object that is subject to preset compression angles. Uh, in other words, you are not really getting closer to the object you are bringing in already compressed distant object closer to you, and peering inside those compressed angles at a slightly larger angle. But the perspective doesn't change. That's important to know. Um, there are some changes in aperture and focal length, etc. that do affect the resolution angle, but think of it conceptually as peering deeper into the set perspective or compression angles that are already pre, uh, present at your distance. Your eye is being brought closer into the already, uh, set angles. I can't stress that enough. However, you cannot resolve something less than theta. No matter your magnification, the angles are set. You just effectively bring your eye closer in. So we have, uh, you know, a perspective view of without magnification on the left, and we have a view on the right. Note that the view, it's the object itself did not change. We just kind of brought our eye closer in to those deeper in angles. But the, uh, the height or width of the object or nothing like that changed. We just brought ourselves somewhat closer in. Essentially, the eye is brought into this wider angle where more information exists. It hasn't compressed yet to that, uh, quite acuteness that it reached, um, towards the back. But but the angle itself doesn't change. The two rays are still at the same. Uh, angle coming in. You've just brought your eye into the Maw or into the less compressed aspects of that angle. Um, so if I magnify this out, you see that I essentially just took the inner, uh, inner compressions, and I pulled them out. And that's what you're seeing with magnification. So again, perspective is observer dependent. If you were to change your camera position by getting closer to the object or higher up, you would see more of the object due to a larger angular resolution and alter perspective, but without changing your position. In all cases, no matter what variables we manipulate, magnification does not alter the perspective of the observer, it only magnifies it. If you have lost an angle or have lost an object due to theta, you will not bring it back. Before moving on. As an aside, we can also use angular resolution to see how far something away is, just like we've done. But we can do this with, um, astronomical, um, uh, objects. So take the sun and moon, for example. The moon fits within 29.3 to 34.1 arc minutes of our vision. This is the theta at which the moon is observed to fill or its angular size. And if we don't know the true object diameter, height, or distance, we can gather a range of sizes and distances that scale to one another according to that angle. So in this case, 30 arc minutes um, is 0.5 degrees. We can plug it in to our formula here. And we can see that the ratio of, uh, height to distance must be 0.0087. So the diameter of the object must equal the, uh, distance by a ratio of 0.0087. And I just gave the picture of the hand. And in the upper right there is just you can give a rough, rough estimate of what an angle angular size of an object is by using your hand. Um, your hand as a fist represents about ten degrees. If you use the pinky and the thumb spread out like that, you can gain about two fist widths or 20 degrees. Using three fingers gives you about five degrees. And using your pinky finger is about one degree. So it's just a rough estimate for getting angular size of a distant object. But going back to the moon example, uh, again, we need a ratio of 0.0087 of diameter to distance. So, um, if the moon is a known distance away, then we can calculate its diameter. Or if we know for a fact its diameter, we can calculate how far away it is. So if it's 238,000 miles away, uh, its size is therefore about 2070 miles, which is what we, uh, or what the heliocentric, um, model believes to be the case. However, if the moon were 3000 miles away, its size would be 26 miles in diameter. Because that also works. It's the same ratio. Likewise, if it were 300 miles away, it would be 2.6 miles in diameter. And so on. Interestingly, I think of importance, the sun also takes up the same 30 arc minutes of vision in the sky, and so it too could also represent the same exact sizes. So we will see how the angular size of the sun and moon are observed with perspective later on. But you need to have two objects, one near with a known distance and height, and one far with a known distance or height. To get an accurate angular size or distance of that far object. We don't have that for the sun and moon, so if physical objects, if they are physical objects, they are quite possibly the same size. Um, as a note, uh. Aristarchus in 300 BC. Assume the lunar eclipse was the Earth's shadow, but without the assumption all these sizes and distances, uh, essentially fall apart. Also, because Polaris is roughly 90 degrees above us at the North Pole, we can calculate, uh, we can use it to estimate our latitude if we assume it has a fixed position. So if we use an astrolabe or a sextant and measure the angle between the horizon and Polaris, we will get our latitude on a ball. It should be physically impossible to view Polaris beyond the northern hemisphere, because you will have gone over the curve. They make it vast distances away, so that they can assume the rays come in parallel with a tangent through the equator. No tangent exists in a straight line beyond zero degrees at the equator, though, because again the rays are blocked by a curvature. Polaris, however, can be seen as far away as 23.4 degrees south, which is the Tropic of Capricorn. So to make the math work, they tilted the Earth and the new equator became 66.6 degrees to Polaris. Now suddenly it was okay again for people to see it from 23.4 degrees south, as well as other celestial observations. So again, just in the top right, you can see at 90 degrees at the north, 90 degrees north, Polaris is directly above you. And as you move outward, uh, toward the equator, uh, the angle changes such that where you view it becomes your latitude. And then on the bottom here you can see how the tilt was manufactured, uh, to, uh, allow people at 23.4 degrees south to maintain a parallel ray profile to view Polaris. But an alternative explanation is that Polaris sets at 23.4 degrees south into the lower half of our visual field, which is set by theta, and it happens when we are approximately 7,824.6 miles south. And again, video links in the descriptions for that. So do the math on that. Assuming accurate latitude, each degree of latitude is 69 miles, so that makes Polaris at least 23.4 plus 90 times 69, equal to 7824 miles away. Since it only makes a very brief appearance above the eyeline or horizon at that point. So it also means we can see celestial bodies 7800 miles away. So for a circle of diameter of about 15,600 miles across our visual field. So at the least we know that is the limit to our diameter of vision, across which we can observe celestial objects. And then just keep in mind the 23.4 degrees south is the Tropic of Capricorn. The zero degree equator. Then you have the 23.4 degree north Tropic of Cancer and so forth. You may have noticed the inverted reflection of the sun and the sunset picture we used earlier to describe how the sun would set into our lower horizon line. So this brings us to the inferior mirage and a brief discussion of refraction. So depending on atmospheric conditions and inferior mirage is something is sometimes seen below the horizon or eyeline. This represents the light bouncing upward to strike your eye inverted as a reflection. So if conditions warrant the viewing angle of the information below, your eye line or the bottom 50% of your vision is receiving light from an object near the resolution limit or horizon. Like skipping a stone, the light received by your eye across distant water is hitting an atmospheric layer and bouncing up to meet your eye. Instead of you seeing the light radiating from the water in a straight line. So you never see the water that's there. Instead, you see the light above it inverted and sent to your eye. The light that was supposed to reach you could not penetrate the layer, so instead you view light that met a critical angle of refraction from above and bounced to you inverted. So from a perspective view, there still exists water within this inverted zone. You're just resolving the light bouncing off the sky above, rather than the light coming off that far away water. If you were to increase your observer height, this layer shrinks away because the light from the farther out water now has a more direct line to your eye through the medium, instead of the critical angle that it meets with refraction. So I tried to depict that here on the lower right with this green vapor layer, by taking some of the yellow, uh, rays that were coming off the water, the rays that were the most shallow and were trying to reach your eye. Uh, they refracted away. And I didn't depict those here, but they met, uh, a critical angle of sorts, and they reflected or refracted away from your eye. They never met your eye. But there are, uh, uh, angles above that. Hit this layer and refract to your eye inverted. And that is essentially what you're seeing. You're seeing the reflected sky or mirrored sky above, but your horizon line is still your eye line. There is still, uh, potentially water within that perspective. If you didn't have this vapor line or vapor layer present. In other words, if I take away the green layer on the right, I would see what I see on the left. So on the left side here I am assuming that I have met theta for both above and below my line, and I have, uh, converged those images together, and you see that the sun is the sun is setting a little bit and the water is setting a little bit as well into the horizon line. Um, and they are meeting without any reflection at that point. Once I add in this layer some of the light rays that are most shallow and most distant coming off that water, uh, refract and uh, uh, are not ever sent to my eye. And instead there are some angles from above that are able to refract and uh, uh, meet my eye inverted. And so I see again that mirrored sky. So to account for some distant observations that should not be seen on a globe. Some will claim that in addition to the inverted images below, the upright versions above are images or mirages as well. In other words, the inversion below is not is a mirage. But not only that, the object above the inversion is also a mirage, and they're both fake. What we're seeing a Fata morgana effect or other mirage effects, but you can video the same object, uh, linearly, uh, diminishing as it would or as it should without distortion as it travels. So you can place a straight line along that image. There is no distortion of the image or movement of the image disappearing, reappearing, um, uh, moving up and down like it would with, uh, uh, typical refraction phenomenon. That is the image itself. It's just moving further away. So here's an example of a boat coming into view. You can see the inferior mirage or eye reflection occurring at various points along the way as the boat travels away. Again, video link in the description. Keep in mind that the water exists, uh, within that zone, within that yellow zone that I've marked out with that arrow, water still exists there. The purple line represents the possible I've just placed possible because I'm not sure within this, uh, video, but, um, the horizon would be above that layer. And if that layer were to disappear, you would see water behind the boat as it travels away. So if our sun was setting, if we go down to the bottom example, we pull out the boat. If our sun was setting in this boat scenario, it would set at wherever the true horizon was located, which is depicted, um, uh, to the right a little bit. So I just placed that other picture just as a illustration of, uh, where an object would possibly set, um, if it were, if we, if it were within this boat example. And this is another way you can use you can use objects that are like the sun or moon that are setting. They would determine your true horizon line at that point because they are so high and distant. When they do set, that is your, uh, your lower limit of your visual field and is not distorted. Okay. So again, just like these are actual vehicles, those are actually the boats that are going off into the distance. So the same phenomenon is occurring here. You see this, uh, this RV coming over the horizon here you can see the inversion of the distant mountains somewhat and the reflection below. And that when you see that, uh, reflection, the middle of it essentially is the horizon line for the observer. Um, and you can see the bottom of the inferior mirage where it intersects the, um, the camper, uh, where it starts to invert. That's just, again, that's that's a false, uh, horizon line. That's just the inferior reflection point of that object. But again, just like in the boat example on the lower left, water still exists all the way up until potentially that purple line. Let's say you're just seeing the reflected sky above. So if you take what's above that purple line and you just invert it, that's essentially what you're seeing below the purple line. But it's not your horizon. Your horizon is is the purple line. I hope that makes sense. Your horizon is not that lower water line where it the water starts to become apparent. Okay, again, just like this is an actual boat up here in the top image, going off into the distance with an inverted image below. Um, this is an actual dog. So again, actual boats, actual cars, actual dog. These are not, uh, inversions upon inversions of images. So when looking at it, long distance photography arguments over whether the bottoms of objects should be seen should take a back seat to observing the tops, because there is simply too much distortion near sea level to make sweeping claims, unless refraction is at a bare minimum. As an example, if we now ripple um, uh, some water with a ruler in it, you'll see the lines expanding contract of the ruler. Concentrate between the one and two centimeters marks and you'll see it expand and contract. So sometimes the ruler is longer, sometimes it's shorter, sometimes the numbers are closer together and sometimes they're further apart. So if we make our water waves a lens and the ruler a building, you can see how the floors may distort on a building when you're viewing it over long distance. So you can see how the waves distort the lines, and you can see how they compress and expand. Um, uh, as they go mostly, uh, mostly compressing and then coming back to baseline. And again, you can also see this effect on a time lapse. Uh, time lapses are great ways to view these refraction, uh, phenomenon. Just look at the buildings, kind of raise up slightly, go back down again, compress, expand. How the water line goes up a little bit, goes down a little bit. The mountain in the back in the background goes down. It comes back up slightly, but mostly, uh, is down and stable. Uh, you see a little bit of inversions occurring. Um. So you can see a lot of effects going on just within this single time lapse of a not too distant shoreline. It's almost as if you are looking through rippling water to to see these effects. So I've kind of depicted that here on the diagram. It's almost as if you're looking through these wavy, uh, phenomenon or wavy atmospheric conditions to see an object with various lensing effects. So sometimes you view water, uh, meet the eyeline correctly. Sometimes that water meets it. Uh, somewhat correct or without a terrible amount of distortion. And sometimes you see an inverted reflection instead. Sometimes refraction pulls the distant objects lower, and it seems sometimes it's higher. Uh, you can see there's very low compression of the, um, buildings in, in one of the views, and you can see a much higher rate of compression. Those buildings are squished together much, much more, uh, in other views, sometimes there's inversions, sometimes there's not. You can also see it in this example below. Again, video links in the description of this is of some distant oil rigs. There is no change in position of the height of the observer, but the distant rig is brought down and the near water is brought higher. This is upward refraction above our eyeline and downward refraction below it. Essentially, like the less dense, warmer air above the sea is acting like a concave a concave lens of sorts, less dense or less refraction in the middle, similar to eyeglasses or binoculars or telescopes, things like that. So we are getting a lensing or refraction effect through many different layers of vapor or Si aerosol that are subject to air currents, temperature, humidity and pressure gradients. So to see the effect of this lensing, let's take another look at the skunk Bay time lapse stills and just, uh, outline or mark out some of the landmarks. So we'll mark a, uh, a building through and we'll take some still shots of the time lapse. We'll mark the building as the lighthouse with the yellow arrow, its position, and we'll mark the water line where it meets the shore along that distant mountain range with a blue line. And we'll mark the top of the mountain with a black line. And you can just see how those change within different atmospheric effects along the way. So the lighthouse changes apparent size. It changes apparent position somewhat. The mountain, uh, seems to get a little bit higher. And then it also seems to get a little bit lower. Um, and the water line seems to get a little bit higher and a little bit lower. There's just many different effects going on here. So again, as you can see, the mountains are higher than lower. The lighthouse is taller than shorter. The water is higher than lower. Sometimes there is concave lensing where the mountain is lower and the water is higher. Sometimes it is convex lending and the mountain is higher. In the water is lower. Sometimes there is uniform upward refraction and both are lower. Sometimes there is uniform downward refraction and both are higher. The effect is variable, but keep in mind that while refraction is highly variable, diffraction is relatively stable. What is lost to the observer from diffraction is wavelength and aperture dependent. So keep that in mind. While refraction effects can be highly varied, diffraction is much more constant and consistent. Um, what we've tried to depict, though, here at the bottom is that you're it does seem like you're essentially seeing it through different waving phenomenon of this, of this atmospheric effects. So depending on the air currents and gradients and pressure gradients and, uh, see aerosol levels and everything that you're looking through and temperatures and so forth, you're going to get various refraction phenomenon. That's just there's just no way around it. Um, but keep in mind, refraction over water is almost always refracted upward. The specific heat of large bodies of water keep the air above the surface relatively warm and stable in comparison to the colder air, uh, higher up. Uh, but again, video link in the description. So for almost all observations across vast distances, the warm air below and the colder air above create upward refraction of an object and the objects are brought downward from the top. So again, you will most often see buildings appear lower than they actually are. And you can see the diagram of what that downward refraction would be like with an inverted prism essentially. And again, uh, laser. Uh uh, most all laser experiments show above large bodies of water show upward refraction of, uh, laser light. And, uh, you can see somewhat the effect of a laser light being shot through a cooler, denser air medium above with a warmer, uh, less dense air medium, uh, below. And again, um, you can see how it refracts upward. What this means is that because our eye sees things in straight lines and the angle is now coming in, uh, somewhat different again, we assume the straight line which brings the building lower. So if you see the yellow line here, uh, that's, that's how the ray is going in. And then getting refracted through the prism to meet our eye. And so what our eye then in return thinks is along that black dotted line. It thinks that that's where the actual object came from. And so it, it essentially brings the entire object down lower. Um, also of note, you get more refraction with colder temperatures and lower wavelengths, so something to keep in mind. So again, with these effects in mind, we can take a look at the Chicago skyline. These are filmed 60 miles away from Chicago. Uh, in other words, they're they're filmed across Chicago from a place called Saint Joseph's, typically, uh, along, uh, on the other side of Lake Michigan, 60 miles away. And you can see the same effects observed during the skunk Bay, uh, time lapse footage. So these buildings are relatively stable? Uh, somewhat, but they are brought lower by a combination of diffraction limit and the warm air over Lake Michigan. So you both again, both of these images are taken in the spring, but you can kind of see somewhat of the wavy, um, appearance of the buildings, just like we, we saw in the, uh, skunk Bay, uh, footage. But again, these buildings are lost by a combination of this upward refraction of, of this warm air layer over the water, bringing these buildings somewhat lower, uh, down, just like the oil rigs. Um, and a angular resolution limit that is cutting off parts of the lower aspects of the buildings. So let's look at these now. You can see the. Almost essentially the skyline just come to life along that lower edge. We'll take a look at this one here. Again, you can see much more of the landscape of Chicago along this one. So before leaving refraction, there is an interesting experiment used to detect ether called airy's failure that used an intersection between motion, parallax and refraction. So I wanted to mention it here. Um, if a stationary telescope points at a stationary star, the telescope will never require modification to observe the star. If either the star or the telescope are moving, an adjustment must be made throughout that movement to maintain the observation. So it was noted in 1729 that a telescope had to be tilted five degrees to get a star in the center of the telescope. Since the Earth was assumed to move and the sky was stationary, this was presumed to be the reason for the tilt of the telescope, so airy decided to test this idea. Uh, and the two presumed reasons for their observed phenomenon are below. So given option one, light moves straight down through the ether or atmosphere, moving in unison with the earth. Both are moving together. The light moves down in a straight line. You have tilted the telescope five degrees and as you are moving, the light ray is essentially as it goes lower. You are moving to the right and so it keeps going lower. And you still observe, uh, the light ray hit the very bottom of your telescope as your telescope moves, right. Versus. The other idea was that light moves straight down through an ether or atmosphere, and it moves independently of the Earth's movement. So in this case here, the light ray is now, uh, moving independently across a stationary, uh, tilted telescope. Or it could be that they're both moving, but that they're moving independent of one another. So again, we have a potentially stationary tilted telescope keeps moving or descending. Uh, it keeps the moving or descending light in the center. Those are essentially our two options between, uh, the two. So to test these options, they filled the telescope. The tilted telescope with water, which would now refract the incoming light. If the light hit the water at an angle, it would bend. And as the telescope kept moving, the star would be lost from the center view. If the ether and the earth moved independently, you would lose the object with greater tilt. So conversely, if the ether and Earth moved independently in unison, greater tilt would allow you to see the object in its entirety again. So what he found was that the ether and the earth moved independently from one another. Uh, so as he tilted the, uh, telescope, he saw, uh, absolutely nothing. And he just essentially the water, uh, had no effect, and the light still hit the observer. So it was deemed a failure to detect ether within the moving Earth as independent or excuse me, as as dependent and moving with the Earth. He never mentioned the more interesting aspect of this experiment, though, that the Earth was stationary and that the atmosphere and ether moved. Uh, later experiments by Michelson-Morley, Michelson, Gael, Signac, and others in the late 1800s and early 1900s continued to prove that the Earth was stationary with various different experiments, but they maintained the assumption that the Earth moved, and they modified their findings to say that the ether must not exist. And this required light to become fixed and matter to expand and contract to meet it, and shortly afterward relativity was born. After a Lorentz contraction with an e equals MC squared equation that everyone is so fond of. These ideas then fed off one another with equations that stated something must exist, and therefore it did exist. Uh, shortly thereafter, and we became lost in a theoretical, timeless dimensionless multiverse. All of it based on the faulty assumption and presupposition that the Earth rotates. Before moving on, let's do our first calculation of Earth radius using the optics of the human eye. For this one, we'll use the vertical field of view, which ranges from 130 to 135 degrees. It's slightly better below the eye because or than above because of the superior orbital rim, but the angle represents about 65 to 68 degrees above your eye line and 65 to 68 degrees below it. For a refresher on Polaris, let's return to our diagram and explain the distance relationship. Recall that Polaris, which is fixed in the sky, moves higher about one degree every 70 miles or so as we go north. They chose 69 miles. The alternative explanation is that if I move 70 miles closer to Polaris, it rises one degree within my perspective. But assuming that the Earth is curved and Polaris is fixed, if a person walks 23.4 degrees south to 90 degrees north, they would cover about 7,824.6 miles on a sphere. So if this represents 23.4 plus 90 or 113.4 degrees, then 0 to 90 degrees north would be about 6210 miles, or 69 times 90. So you can see we've just taken this globe representation. They've tilted it to a fixed, unmoving Polaris, thinking themselves that they are moving across the Earth to, uh, to keep the fixed object within view going one degree every 69 miles. If we lift the arc off of the Earth, and we can kind of represent it here along a straight line, and we see that as we move south, uh, we can using a sextant and comparing the angle to the star, uh, with the horizon, we can get our latitude. We've already gone over that. But as we'll see, what's really happening is that instead of your vertical field of view being 65 to 68 degrees, they made it 66.6 degrees and failed to to take into consideration an additional 23.4 degrees of vision that occurs within perspective, intentional or not. Without this additional distance, they had to tilt the Earth. So we'll explain that a little bit here. Looking at our corridor our vertical field of view limit is here. Um, over on the left, uh, you can see the, uh, you can see how the angles diverge and converge from the light source. And they are converging back on our lens. So think of this limit, um, as a limit of lens convergence. So this this is limits the angle at which we can create an image above us or below us, looking straight on at the horizon. So when we collapse our corridor, um, around this angle into our perspective, it becomes the closest angle to our eye. In other words, it's the closest angle with which we can view those paint stripes, essentially. We can drop a vertical line from the points where these two angles reach the ceiling and the floor. And again, it limits the distance along the floor or ceiling that something can be viewed above or below us. But again, on a straight line view, we can still see an additional 23.4 degrees further. So again, our paint, our paint strips that are attached to the ceiling or floor may uh, may limit there. But looking straight on at an object, we still get an additional 23.4 degrees of vision. So if we're viewing a distant star, however, we're estimating that its distance to our eye. So this is essentially our eye line. If the object is setting far away. Right. So it is taking into consideration that 23.4. So if I'm using a sextant to measure something and comparing the horizon and the object, I'm evaluating it according to these vertical angles until 65 to 68 degrees, which is a maximum. After this point you must tilt your vision upward to view the object. But if I don't take into account the vertical field of view, I do not take this additional 23.4 into account. So I believe something is 90 degrees away when it's actually 90 plus 23.4 degrees away. If we use 66.6 as our vertical field of view above, our eye line and object along the ceiling would be a certain distance plus 23.4 degrees. Again, this represents the other angle of that right triangle. So this distance is what is ignored when viewing Polaris from 23.4 degrees south. So we'll assume the vertical field of view as 133.2, which is just 66.6 degrees split above your eyeline, 66.6 below. Um, we'll recall that when something is high up, we are essentially rotating these angles down within our perspective to our eye line as they approach theta and set into the lower eyeline. Again, once theta is met, that sets the lower half of our visual field, and whatever is left above our eye line becomes the top 50%. So looking at our diagram, we can reverse it to fit within our corridor at the maximum height and distance at which it can be observed. Over here on the right. So we'll just flip our diagram, flip the star and the eye. And the height there represents the furthest distance and height it can be. So again when rotated down this is the furthest point along our horizon, uh that we can see. As discussed the line represents 90 degrees plus 23.4. So from the zero point to 23.4 represents uh about 1614 miles, 69 times 23.4. And from the 23.4 mark, um, to our horizon line is the 90 degree, is the, uh, basically the 90 line to Polaris or 90 times 69, which is 6210 miles. And adding them up gets us our 7,824.6 miles. As you can see on the lower left, we've kind of outlined how those distance relationships work themselves out. The red line represents that vertical line dropped along, uh. Uh, from that 66.6 degree uh, triangle along 23.4 degrees, dropping it down perpendicular to the eyeline. So if I believe I'm on a globe and 7,824.6 miles represents an arc length, uh, from 23.4 degrees to 90 degrees and 6210 represents an arc from 0 to 90. Then if I multiply my quarter arc of 0 to 90 by four, I get a circle with a circumference of 24,840 miles and a radius of 3953 miles. Keep in mind the radius of Earth is 24,900 miles and a radius of 3963 miles. So again, what they think is happening is up here on the right. What is really happening is that they are not taking into consideration that additional 23.4 degrees that is dropped from the vertical field of view to our eyeline. So this is one way they use the optics of your eye to tell you that you live on a sphere. Of note, the implication when viewed correctly within perspective, is that when viewing celestial objects, if we move 70 or 69 miles closer to something, our horizon opens up about one degree, or it moves along one degree within perspective. Put another way, if we stay fixed, we see an atmosphere with a radius of roughly 7800 miles and a diameter of 15,600 miles. Lastly with the inferior mirage effect. Understood. Let's look at another example of diffraction limit. The lid in this example, in a warehouse is marked at one inch intervals. The observer height is 1.5in. The pictures are taken at 20 foot intervals, so this is a fixed height changing distance example as opposed to a fixed distance changing height. Example again video link in the description. So taking still shots of the video capture here or the picture capture, we have 100ft, 140ft, 180ft, and so on all the way to 460ft. And you can pull these objects out and compare them to one another between the 460ft and 100ft, which there is very little to no angular resolution loss at 100ft. And so if you overlap the two with the scaled paper of one inch intervals in between them. And mark out the horizon line as the point of the inferior mirroring effect, then you can see that you have lost approximately 1.75in, um, at 465ft. And again, you can use this to then calculate theta for this particular um, wavelength diameter aperture uh camera. So we'll do that here. So, uh. Again, this 1.75in is lost at 460ft, so we plug in 1.75in as our height, which is 0.1458ft across a distance of 460ft. Using our same diagram as we've done before, solving for theta gives us a very similar angle to what we've gotten before, which is 0.018 degrees, very close to our 0.02 degree, uh, arc minute, one arc minute human eye, uh resolution limit. So using the loss to calculate theta for this camera within these conditions, I can now calculate the amount of loss over any distance. Alternatively, I can calculate the amount that is resolved with a change in height, just like we did before. So plugging in at 260ft, I would expect a loss of about one inch. At 380ft, I would expect a loss of 1.43in and so on. Uh, if theta is held constant across any similar triangle, we can restate our formula as h one over h two equals d one over d two. Right. And those those ratios then could correspond to one another across tangent theta uh as h one over d one equal to h two over d two. So if you know the distance to an object and the amount that is lost across data, you can calculate when the object would be completely lost. So for us, our 11.5in height lid completely disappears over the horizon at a distance of 3050ft, or 0.57 half a mile, give or take. Alternatively, over a distance d of one mile 5280ft, we would lose a height of 1.65ft, or 20in. Of note, if this lid were instead a six foot man, he would disappear at 3.6 miles within these wavelength and aperture conditions. Again, this is an optical phenomenon is not physical curvature. The lower horizon limit has been met. The compression rate has been met below the eyeliner, below the aperture, or below the the eyeline of the camera in this case. And so the upper aspect with distance is going to set into that line. As another example, you can see the effect more clearly here. The lid is marked at one inch increments and the photographed one point, and is then photographed 1.5in off the ground at various distances away. So some of the pics were downloaded at different levels of zoom, but you can see the effect on the lid. It disappears into the floor and the whole lid is brought downward. You can see the lines on the lid gradually merge with the floor, and I've outlined those markings and placed them against the purple line. Um, and you can see that you just you start to lose them all sequentially, uh, as with greater distance. So let's summarize where we've been up to this point. I want to impart to you that the loss of information at our horizon is an optical phenomenon. The ground is brought to eye level. Our eyes are not brought to ground level. Distant objects disappear by a combination of the angular resolution limit of the observer and upward refraction phenomenon. So option one again the ground is brought to your eye level. The person disappears from perspective by walking straight away and vanishing bottom first from things getting smaller with distance and an angular resolution limit. And here's some of the examples that we've gone through. Uh, to this point. We have our lid disappearing into the floor. Uh, we have some calculations that we can use to determine our horizon distance, uh, and or loss rate. Um. Uh, with height change. Let's now turn to our second option. So option two was we had the AI is brought to ground level. The vanishing person is represented orthographically without regard to perspective, and your focus continually shifts downward to observe them going over a curved surface and vanishing at the same rate. It just so happens as the angular resolution limit of the human eye, and we will show that momentarily. So again within this diagram our eyeline shifts downward. It must shift downward to observe an object physically going over curvature. Perspective effects and angular resolution are ignored. And again putting it together so you can see them side by side. Uh, the loss of information at our horizon, just to restate our argument, is either an optical phenomenon or a physical one. If it's physical, they disappear by walking over the horizon. And if it's optical, people disappear by angular resolution limit of the observer. So with these options in mind, we will now use the angular resolution limit of the human eye to calculate the radius of Earth. The first step. The first step in doing this is to assume a curve, and that the lower 50% of our vision represent a physical blockage of our vision. So we need to translate our linear perspective into an exponential one. So again, the angular resolution limit of the human eye is one arc minute or 0.02 to 0.03 degrees. We have to get a more precise value though, or the range will be too high for the radius of Earth. So we need to plug in some good numbers for distance. Since we're seeing boats go over the horizon, we'll use a small pupil diameter because we're focusing at far distance. So again the diameter of the pupil ranges from 2 to 4mm in bright light and 4 to 8mm in dark. Recall that refraction effects are highly varied. So we need a relatively non refracted wavelength to get a more consistent value for a distant object. So orange red wavelengths are less resolved than blue, but they are less subject to the highly variable refraction effects and atmospheric scatter that we witnessed before. Um, recall that refraction and diffraction are different. One is the change in angle through a new medium, one is the change in angle from passage through an aperture. So the wavelength of light ranges from 400 nanometers of the purple range to 700 nanometers of red. We want to use the more orange red, uh, later wavelengths. Since we're looking at objects far away, we'll use an orange red wavelength of 666 nanometers. The pupil narrows when it focuses on distant objects, like we said. So we'll use a pupil diameter of very low of 2.16mm. So we can solve for theta using a predictable occult preoccupation with numerology. And the number nine, uh, such that theta in radians is equal to 1.22 wavelength divided by diameter, which is 666 nanometers divided by 2.16mm. That gives us our, uh, number in radians, and which is also, uh, adding up to nine as we'll go through. And that gives us a angular resolution, uh, theta of 0.0216 degrees. So again I chose these numbers on purpose, the numerology of sixes and nines as six, six, six, which adds up to nine and two point or 216, which also adds up to nine and 3.762, which also adds up to nine. Ah. Uh, on purpose. Very intentional. The subject is vast, but it is there. Mark, just like 24.3 degrees is nine and 66.6 degrees is nine, 432Hz is nine. The sacred geometry of sound and cymatics the seconds, minutes, and hours all are increments of three, six, and nine. There are Fibonacci pairings, flowers and trees of life, and numerous other examples of numerology that seem to stem from Pythagorean and Ptolemaic schools alchemy, Hermeticism, and Babylonian and Egyptian Greek paganism. In essence, the six is man's number, the nine is their god number, and the six six, six is how man becomes God. As six, six, six is equal to 18, which is equal to nine. I won't go into that too much at this point. I just know that the numbers that I chose, uh, were, uh, purposeful. So we calculated theta as equal to zero point uh uh 0216 degrees for long distance vision and a pupil diameter at the lower limit. So we can now use this angle to see how far the average six foot person can see. Uh, just like we did with our object, um, examples before. So distance is equal to the height divided by tangent theta, which is now equal to six feet divided by tangent 0.0216 gives us three miles. With theta and our max horizon distance known at a six foot elevation, we can solve for the amount of information loss at that distance. So again, we flip the triangle back around and this person at this height would lose six feet. So under optimal conditions with a minimum pupil diameter of 2.16mm for long distance viewing and a wavelength of 666 nanometers, we can see that when looking three miles away at a six foot observer height, anything smaller than six feet below the horizon cannot be observed. We now have the two triangles with which we can construct a curve. We can fit the height and drop triangles together as we look down across the curve to view a distant object disappearing over it. So again, the first one is the distance that we calculated for a six foot person using theta at 0.0216 degrees. And the second one is the loss of information that that person would, uh, uh, be granted across that three mile distance. And you can then use those triangles together to get your rate of curvature. For a six foot person observing someone go over the curve. So as an aside, if we extend the lines, you can see that we underestimate the loss at this pupil diameter and wavelength nearby and overestimate the loss using curvature with greater distance. So again, if I assume a curve now that this green person essentially is walking over a curve, um, you see that the lines don't really match up between the triangle. The straight blue line and the brown line do not match up. Uh, the curvature rate is underestimating it, uh, nearby. And it is grossly overestimating it. Uh, far away. So, for example, at 100 miles, we expect to lose roughly 184ft by angular resolution at .02 degrees, but with curvature at 100 miles, we're losing 6660ft. So the takeaway is that with distance, diffraction plays less of a role than refraction does. That's the takeaway here. Uh, because we do see, uh, the bottoms of things, uh, lost at a greater rate than simply 184ft at 100 miles. By extension, the use of infrared camera technology is very useful over long distances, since it is more prone to diffraction issues but experiences less refraction at the same time. So, um, because refraction does play a decent size role, uh, that the infrared camera technology is proving to be, uh, uh, a very good tool to view these distant objects, which, again, do not have anywhere near the expected rate of loss, uh, such as 600 and, you know, 6000ft of loss at 100 miles. It just doesn't exist. So our rate of curvature is six feet lost at three miles from a six foot observer height. We need to approximate the angular resolution limit line within our curve as close as possible. We only have two crossover points, one at zero miles and one at three miles. We know that the size change must be exponential within perspective in order to get a curve to develop out of. It can't be linear, so we will solve for the Earth's radius by knowing the arc length and arc angle. You can see here. We've just taken our two triangles, placed them against the curve. For a person walking over at that rate of six foot loss over three miles. And we have then extrapolated it along a greater circle. So the observer would assume they are watching a person descend from their point of view across two theta at three miles. The first theta, 0.0216 degrees, would represent the observer height at six feet until reaching zero feet, and the second theta for the drop from 0ft to -6ft. So we can therefore take our arc angle of two theta across an arc length of three miles. Or we can take the drop rate of 12ft across three miles. Either will give us R for the greater circle. We can now solve for the radius of Earth from the angular resolution limit of the eye, as our hour segment here with R along the radius, or along the one aspect of the triangle, and three miles as r, uh, opposite end, and the angle being across two theta, which is 0.0432 degrees, that will give us one number. And the alternative is to use the ratio of 12 foot over three miles is equal to three miles over R. Both will give us the numbers listed below. So we get for one number 3978 miles, and for the other one 3959 miles, giving us an average of 3969 miles. So for our circle, based on the angular resolution limit of the human eye, we can calculate a radius of 3969 miles. And the circumference then comes out to 24,938 miles. And coincidentally or not, the Earth's radius is said to be 3963 miles with a circumference of 24,900 miles. We can then calculate the drop rate and horizon gain that comports to a curve with the radius of 3963 miles by pulling out our triangles as below. So if I now deconstruct these triangles that we used to get the curvature rate, we can get either a horizon gain or a height loss or drop rate. So recall that within option one of the angular resolution horizon gain is nothing more than the expansion of the lower eyeline resolution angles or theta expansion, and the height loss or drop is nothing more than the compression of the lower eyeline resolution angles or the theta compression. So we can then extrapolate these two triangles, uh, to a greater circle again and calculate out for a. Circle what we would have as a height loss or drop, which is again just the theta compression or the horizon gain using trigonometry, which is again the theta expansion. So for the circle here with this particular rate, to create a circle you need sine theta is equal to where theta is that arc angle there between your the distance over which you're observing something drop. It's equal to the distance over which you're observing it drop divided by R the radius of Earth which yields that angle. And then you're going to solve for the height of that drop such that the height is going to equal the, uh, radius minus the radius times the cosine theta. And you can see from the trigonometry there how you get those angles from uh, from those, um, from those arc lengths and so forth. Um, you can alternatively get a horizon gain or theta expansion formula pretty simply by just using a right triangle. So in this right triangle example you're going to have um r plus a height of u sitting above the surface of this circle. Uh squared is going to equal r squared plus d squared. You can see just the edges of the right triangle there. And you can then solve uh for uh the distance d um the drop h squared is since it's small, you can just subtract it from r squared from both sides. And then d becomes the square root of two hr or distance in miles is equal to 1.22 times the square root of height in feet. So if you just plug in your height in feet above the surface of the circle, you will get the distance d across which you can see. And you can see the exact same formula here on the Wikipedia horizon example, saying that the average person, say about six foot can see about three miles. Which is again just angular resolution. So we can also derive a quick calculation formula any number of ways we need to determine the exponential rate of loss across a change in distance. So in other words, we need to know what the loss per mile is with each additional mile or the x loss of feet per mile squared. We can then model this rate of loss two ways. One ignores perspective and models a standard height that experiences an exponential drop rate, or attempts to incorporate perspective and models a standard drop rate across an exponential size change. So here we have our two possible ways to model this. But the same equation. You have a standard drop rate x across an exponential height change, which you could then place into perspective to view something getting smaller with distance and parts of it getting cut off with that distance. So height is equal to x some constant, uh, that we are uh, it's going to be some unknown standard rate of loss, uh, across one over the distance squared, uh, to give us our exponential height change. Uh, and in the other way we're going to use the same formula, but we're going to be using X here as our unknown exponential rate of loss across a standard height. So again on the first example, we have an unknown standard rate of loss across an exponential height. And on the second one we have an unknown exponential rate of loss across a standard height. Both give us the same information. If you need to pause and digest some of these, please do so. Either way, we need to find the drop rate x. So we'll use the standard drop across an exponential size change in order to view this person in perspective. So if we believe the rate of loss is six feet for every three miles, starting at zero feet in zero miles, which are again our two matching points between linear angular resolution loss and our exponential curvature loss. We can then solve for x. So remember we need to know what the loss per mile is with each additional mile or x loss of feet per mile squared. So the six foot loss at three miles becomes a six foot loss at three squared miles, or a six foot per nine mile squared loss, or 0.666ft/mi². So again, you can change our, uh, our formula essentially, or the way of looking at this diagram to be a, uh, just substitute X for what we solve for as 0.666ft/mi². And this person has that part of him, uh, essentially removed with that particular distance. So you can see at one mile he has chopped off about eight inches of his legs. At two miles, he's lost about 2.6ft of himself, or about half of his body. At three miles he's lost one full body. He is six foot has disappeared. At four miles he's lost to about two, uh, about 10.6ft of himself. In other words, if we remove 0.666ft from the object across the square of the distance, um, this is what we're doing in order to theta, adjust our height in perspective. You can then just take 0.666ft/mi² and you convert it to in inches. It becomes 8in/mi², which is the common formula that people use. Again, this is not terribly accurate. It's only accurate up to about 100 miles as well as we'll see because it works out to a parabola. It's not a circle. We can place this person into a perspective view and an attempt to match our reality like we talked about. But this often fails to match what we observe because the math was used to fit a false assumption. In other words, keep in mind these formulas are not accurate, but I wanted to show the derivation again. The truth of the world is that it is linear. So again the loss per mile 0.666 lost below the knees. Then we have 2.6 loss below the waist. Then about six foot loss which is the whole body. And then about 10.6 loss which is almost two bodies. And our inverse square law adjustment here you're going to be at 1 to 1 at one mile, 1 in 4 at two miles, one over three squared at three miles and one over four squared, one over 16 at four miles, and so forth. So you can see the conversion there going from right to left. And then you can just place that person within the horizon line and you see what you are going to get chopped off at various distances. So, uh, our one person here is going to get his legs chopped off at, uh, one mile. He's going to get half of his body chopped off at two miles. He's going to lose. We're going to lose an entire person at, uh, three miles. And we're going to lose two entire people, or almost two entire people at four miles. It's just going to keep going, uh, from there. So I just took the picture on the right, and you just chop off a portion of it and you move it into perspective. That's all that's being done. It's an exponential size change with a standard set loss set against our perspective view. And this is somewhat of our theta adjusted pear pear parabola or parabolic loss. Um. So it can be used to estimate a curve again, but it's only good to a certain point. It works up to about 100 miles, as you can see, and then it just starts to fall away. You cannot make that come back into a circle. So it has a slight overrepresentation of loss, but then greatly underestimates the circle as it goes on. So again, they're they're the two curvature formulas are roughly similar to 100 miles. And then they start to break down. Um, whereas our linear loss you can see that what we actually lose within our linear world is way up there. Uh, so it's it's much more, uh, of a dramatic loss that you have to have in order to live on a curved surface. You have to have these high rates of curvature and loss of an object. So what that means is, if you do see an object and you see more of it than you should, it should bring into question that this is a physical phenomenon, that you are losing this object physically behind curve. We already have the quick formula for height gain according to the circle model, and the assumption that we look down across a bulge of Earth to see distant objects. But the distance across the height gain is simply the inverse scenario of the height. Drop across a distance, so we can invert the drop rate of 0.666ft/mi² as the height gain and horizon distance gain. So the parabolic height loss or drop is then just inverted, uh, to become the parabolic horizon gain. So our height equal to 0.66 of a drop rate across the mile squared, uh, is then inverted, the H becomes one over H. Uh, so then one over H is equal to that same rate. And we can then solve for distance and it becomes 1.22 times the square root of H like we derived earlier. Uh, and they are again very similar in that respect. So the parabolic horizon gain is much more, um, accurate to a circle. But again, the drop formula, when plotted out and compared with the circle, works very well up until 100 miles. And after that that aspect of the formula breaks down. So before closing, let's look at the Chicago skyline one more time. This time lapse is from a dune at, uh, very, uh uh, close to the water. Grand Mere State Park, uh, Stevensville, Michigan, approximately 57 miles away. Uh, and again, video link in the description. Keep in mind that the Willis Tower is, uh, 1748ft high. If you plugging it in, plug it into any Earth curve calculator of your choice. You see that the Chicago, uh uh, the entire, uh, all the buildings of Chicago should be hidden behind 2000ft worth of curvature. Physical curvature. Uh, you should not be able to see them. Just like the skunk Bay footage, the same refraction and mirage effects occur. But again, the image is relatively stable, as if you were just seeing it through wavy water. Not that you are bringing this entire object or entire amount of imagery up behind a curve. Uh, nor that it is compressing and moving back down again. At that rate, it's just not happening. Um, it is not brought up around the curve from downward refraction over water just doesn't occur. If the horizon were a physical bulge, we shouldn't see these buildings at all. They should be lost. So recall that the bottom is lost from angular resolution limit, uh, accounting for roughly at this distance about 110ft at 60 miles. And the remaining roughly 300ft that seems to be lost is from upward refraction across the warmer surface air over the water. And let's take a look at it now. And you have to ask yourself if this is the Chicago skyline, or you believe that this is, uh, all this entire thing is a mirage. Again. To me, that's the Chicago skyline. Um, and it's waving and moving slightly up and down, just like the skunk Bay footage. But just like skunk Bay, you were looking at those buildings across the water. I am looking at the Chicago skyline across the water. So here's the problem. The explanation for seeing these buildings is that they aren't actually there. So again, they're going to tell you that that this is not actually what you're looking at. Uh, so he believes that this is some one time mirage inversion event. But again, there are several time lapses. Then you can see this Chicago is always visible essentially the same way. It doesn't go down below the curve. And then it's brought back rapidly upward. Um, uh it's stable. It's just slightly moving up and down here and there and expanding and compressing here and there. But it's just like all the other phenomenon that we've witnessed with closer objects. So again, the reason is not because of a mirage. The reason that we can see it is because this is the Chicago skyline. This until I found this photo from Grand Mere State Park. This is from Joshua Nowicki, and what you're seeing here is a mirage. We typically would not be able to see this from the Lake Michigan shore. We talked about this last night. Conditions are right on the lake that we're actually seeing a mirage of the Chicago skyline. Very interesting. Here's what's happening. This is a good example of a superior mirage. So Joshua was on the Lake Michigan shore. He was looking towards the West, and Chicago's beyond the horizon should not be able to see it. However, with the right conditions, we have an inversion. We have cold air near the cold lake water and some relatively warmer air above it. This will bend the image of that skyline back towards the viewer, and so typically we would not be able to see this. This image would be viewable from much, much higher in the sky, up in space. So again, the explanation is that you're seeing a Fata morgana or multiple stacks of unstable inverted imagery. Again, I want to impart that the loss of information at our horizon is an optical phenomenon. The ground is brought to eye level. Our eyes are not brought, brought to ground level. Distant objects disappear by a combination of the angular resolution limit of the observer and upward refraction. So again, I'm just showing the earth curve calculator. That must be, uh, that must be enacted in order to, to make these buildings appear. Uh, you have at a six foot observer height and a 60 mile distance, you must have a hidden height of 2166ft. And none of these buildings should be visible. This is not a mirage. That is the Chicago skyline. So again, things get smaller as they move away. Your horizon rises to eye level. That is what is happening. You don't continually shift your eye level downward to see a falling horizon again. Note all the Earth curve calculators require you to be looking down across a horizon to view another object on the other side of the bulge, going up at another angle. The ground rises to your horizon because we see the world in perspective. It is the top and bottom half of our vision. Your eyes were used to deceive you. And you don't live on a ball. Again, cracking these codes is collaborative. So special thanks to Vortex Puppy. Good times for all Chris Vanmeter, Curious Life, Nathan Oakley, and everyone at Flat Earth Debate. Fuck that word Peabrain Wide awake doctor John Dee, taboo conspiracy rant, Flat Earth, Rob Skiba, bml, SB 69, Globe Busters, Jaron ism, Eric Dubay and many more. Again, your horizon is always at eye level, whether at 300ft or three. 30,000ft. There is no curvature. Ships don't go beyond it. Buildings aren't blocked by it, and the sun doesn't set behind it. Water doesn't bend around it. It always finds its level within a container. Our eyes were reverse engineered and we were deceived. So the next question is always why? And to be frank, you'll not fully comprehend the answer. Without a knowledge of the Bible and a knowledge of the occult world in which we live. I always wondered why God would make a universe so filled with occult numerology, but I now know the answer. It's because he didn't. Man did. And those are his numbers, and we don't live in that world. Your eyes were used to create the curvature of Earth, and they did so according to the number of a man, and that number was 666. While it is not the mark taken on the hand or the forehead, uh, it is a mark you are living in days of prophecy fulfillment. It is a mystery in plain sight, and you will bear witness to it if you ask God for guidance. Heliocentrism is sun worship. There is no proof of Earth's rotation or an orbit around the sun. In fact, there are many. To the contrary. We are motionless and light travels in an ether. Weather and wind maps suggest there is no tilt of Earth. It was an attempt to explain the visual phenomenon of the stars. Keep in mind you are supposedly 3 million miles closer to the sun in winter, yet a 23.4 degree tilt is the reason for the seasons. In his 1543 work revolutions, right after giving his diagram of the supposed solar system. Copernicus. References. Hermeticism. In the middle of all sits the sun enthroned in this most beautiful temple. Could we place this luminary in any better position from which he can illuminate the whole at once? He is rightly called the lamp, the mind, the ruler of the universe. Hermes Trismegistus names him the visible god. Sophocles Electra calls him the all seeing. So the sun sits as upon a royal throne, ruling his children, the planets which circle around him. Most of the revered fathers of scientism are alchemists, and Newton was no exception. His translation of the Emerald Tablet of Hermes Thoth is the most revered. There is no proof of gravity. It is a fictional force that bends water around a spinning sphere, holds people upside down, and holds the universe in perfect balance. It's apparently so well understood that we can use it to go to the moon and keep satellites in perfect orbit, yet practical applications are nonexistent. Gravity is just the direction God assigned the vector of down density, buoyancy, and some other physical phenomenon to explain why things float and fall. They packaged their gnosis into an unknowable God and sold scientism as science. They gave gravity supernatural significance and told you that you evolved from space dust. They said you exploded from nothingness and are an insignificant speck within the enormous containerless vacuum of space. They laundered your money and called it a space agency. They put Pluto's face on Pluto in a Saturn cube on Saturn, and it goes unnoticed. It's all around you, and you sit unaware. You live in a mystery religion that they put in plain sight, and you are unwilling to acknowledge it exists. We're all God's children. He set us within a bounded habitation and we are his offspring. The potter's clay denied their maker and thought they knew better. They believe they earned their truth. But we will never fully understand our cosmos unless that knowledge is granted. They turned the world upside down and inverted the role of God. At the highest levels. They believe the biblical God to be some form of a demiurge or an ignorant, shrouded god hidden from the true nature of Sophia Achamoth, their true creator. They believe the real Lightbringer is Lucifer, that the serpent brought knowledge and truth, and his angels are ascended masters who unlocked the creator's secrets so that man can ascend to his godlike potential. They believe their deceptive actions are therefore justified. It is dualism and a series of dialectics where good and evil are sides of the same coin, where the true moral authority is found within yourself, and true ascent is the merging of two in opposition. It is a moral code built on consequentialism and moral relativism, where their ends justify their means, because ultimately you are your own God. Love is love after all, right? You just need to be you. You have to love yourself first. Your happiness is what matters. Just believe in yourself. Do what you will. Love is all that matters. Or rather, do what thou wilt shall be the whole of the law. Love is the law. Love under will. God's commandment to you is to love him and love others. Your self esteem will work itself out. See what we're taught for what it is. It is dilemma and theosophy. The sun could have been scaled to any distance. It was given 93 million miles for a reason. We continue our transhuman efforts as man attempts to climb the Tree of Life and solve the riddle of immortality by breaking open the hermetic alchemical philosopher's stone with knowledge and reason. But those efforts are guarded, and the climb is in vain. The answer to the riddle is that immortality is given freely. It's not earned. We are called to bring lightness to the dark. And I thought when people realize the earth wasn't enclosed, motionless plane, they would fall on their face and repentance. I no longer think that will be the case. Even a new cosmology will be crafted to fit man as God. And the plethora of excuses for globe theory contradictions are evidence to me that some will never give up on the model. We will trudge on like always and replace God with ourselves until our judgment, all prophecy will be fulfilled because truth is not relative. I see two possibilities at this point. One God intended man to climb the tree in search of true self, and by doing so, man becomes like God. Or two. God does not intend for you to climb. God became like man, and by doing so allows yourself to find the true God. We live in a shrouded, gnostic religious world built on an unknowable force. Their unknowable God and biblical significance is everywhere if you seek it. God bless. Acts 1722 through 28. Then Paul stood in the midst of Mars Hill and said, ye men of Athens, I perceive that in all things years too superstitious. For as I passed by and beheld your devotions, I found an altar with this inscription to the unknown God, whom therefore ye ignorantly worship him I declare unto you. God that made the world and all things therein, seeing that he is Lord of heaven and earth, dwelleth not in temples made with hands. Neither is worshiped with men's hands as though he needed any thing, seeing he giveth to all life and breath, and all things. And hath made of one blood all nations of men, for to dwell on the face of the earth, and had determined the times before appointed, and the bounds of their habitation. That they should seek the Lord, if happily, they might feel after him and find him, though he be not far from every one of us. For in him we live and move, and have our being. As certain also as your own poets have said, for we also are his offspring. If you read the Bible with new eyes, you see some important distinctions. I won't go through them here in detail, but I will list them for you for your study. I will hold the screen for a few seconds. Please screenshot if you'd like to bookmark and read the verses. Keep in mind Isaiah 2913 through 16 as the potter's clay. Turn things upside down within globe theory. The Earth's cosmology is known and the Earth is a sphere within the emptiness of space. Earth rotates on an axis and orbits the sun, where the moon reflects the sun and the sun and stars are distant and bigger than Earth. Up and down. Her relative. Within a flat cosmology. The Earth's complete cosmology may be unknowable, but Earth is flat and closed with bounds. The Earth does not move. The sun orbits the Earth. The moon is its own light, and the sun and stars are projections of light smaller than Earth. And up and down are definite. Remember, the Bible interprets the Bible. Repetition requires emphasis. 2 to 3 witnesses are needed, and there are many cosmological constants throughout Scripture. I'll now go through the, uh. Various parts of the, uh, uh, a flat, a possible flat cosmology and where it may have some biblical, uh, roots. And, uh, I would ask that you please study them on your own and to make up your own determinations. And God bless you all. ## Ai Summary of select images: The images depict a visual explanation of how perspective works in relation to the observer's eye level and the phenomenon of objects appearing to disappear or "set" at the horizon. They describe the relationship between an observer's height, the angles of sight, and how these angles affect the perception of objects at different distances. 1. The images illustrate that when an observer lowers their eye height, the angles close to the ground become more compressed, which results in a closer horizon. This means that objects will seem to disappear from the bottom up more quickly than if the observer were at a higher vantage point. 2. They provide a detailed explanation of the concept of angular resolution, which refers to the eye's ability to distinguish between two points. The images discuss the vanishing point, which is the point at which all orthogonal lines converge in a perspective drawing, and how it relates to the horizon line and eye level. 3. Through the use of diagrams, the images demonstrate how the same amount of visual information (e.g., the height of a building) can appear differently compressed or expanded depending on the observer's eye level. As the eye level decreases, the angles of vision below the eye level compress, causing the lower parts of objects to disappear first as the distance increases. 4. The images also address how changes in the observer's position can alter the perceived angles of objects, affecting how far one can see and how the top and bottom parts of distant objects appear to converge at the horizon. 5. Real-world examples, such as observing a tall building or a bottle on the street, are used to illustrate these concepts in a practical context, showing how objects can become visually compressed and seem to vanish from sight due to perspective and angular resolution limits. In summary, the images convey a complex understanding of perspective, detailing how eye level, viewing angles, and distance interact to create the visual phenomenon of disappearing objects, without involving the curvature of the Earth. ![[Attachments/vlcsnap-00549 2.png]] ![[Attachments/vlcsnap-00583 1.png]] ![[Attachments/vlcsnap-00584 1.png]] ![[Attachments/vlcsnap-00586 1.png]] ![[Attachments/vlcsnap-00587 1.png]] ![[Attachments/vlcsnap-00525 2.png]] ![[Attachments/vlcsnap-00544 2.png]] ![[Attachments/vlcsnap-00545 2.png]] ![[Attachments/vlcsnap-00547 2.png]] ![[Attachments/vlcsnap-00548 2.png]] The images provided seem to illustrate concepts related to perspective and the perception of objects at different distances. They likely explain how the height of the observer affects the way objects are seen, particularly how they appear to compress or expand visually due to perspective. They probably contain diagrams showing lines converging towards a vanishing point, which helps to understand how objects that are further away look smaller to the viewer, and the relationship between the observer's eye level and the horizon line. The images may also demonstrate how the angles of sight change as the observer's position moves up or down, altering the perception of the size and shape of objects. For example, as the observer lowers their eye level, the angles of sight below their eye line become more compressed, making the bottom parts of objects disappear before the top parts as they move away. Additionally, there could be examples using everyday scenarios to help visualize these concepts, like looking at a bottle from different heights to observe how the bottom disappears from view as the eye level decreases, or viewing a building from the ground up and noticing how the top floors seem to 'set' into the building due to the angles of sight. These images are likely meant to provide a visual aid in understanding complex visual phenomena related to perspective, angular resolution, and the way our eyes and brain interpret the size and distance of objects in our field of view. ![[Attachments/vlcsnap-00586 2.png]] ![[Attachments/vlcsnap-00525 3.png]] ![[Attachments/vlcsnap-00544 3.png]] ![[Attachments/vlcsnap-00545 3.png]] ![[Attachments/vlcsnap-00547 3.png]] ![[Attachments/vlcsnap-00548 3.png]] ![[Attachments/vlcsnap-00549 3.png]] ![[Attachments/vlcsnap-00583 2.png]] ![[Attachments/vlcsnap-00584 2.png]] ![[Attachments/ezgif.com-animated-gif-maker (32).gif]] ### Card 1: Basic Circumference Formula **Description:** Calculates the approximate circumference of a circle at a specific latitude on Earth. **Formula:** 𝑙=2𝜋𝑅⋅cos⁡(𝜙)l=2πR⋅cos(ϕ) **Parameters:** - 𝑅=6,378.137R=6,378.137 km (Earth's mean radius) - 𝜙ϕ = Latitude in radians (e.g., Tropic of Cancer) --- ### Card 2: Adjusted Formula for Earth's Oblateness **Description:** Adjusts the basic circumference calculation to account for Earth's oblateness, improving accuracy. **Formula:** 𝑙=2𝜋𝑅⋅cos⁡(𝜙)⋅(1−0.00669438sin⁡2(𝜙))−0.5l=2πR⋅cos(ϕ)⋅(1−0.00669438sin2(ϕ))−0.5 **Parameters:** - 𝑅=6,378.137R=6,378.137 km (Earth's mean radius) - 𝜙ϕ = Latitude in radians - Eccentricity factor = 0.00669438 --- ### Card 3: Deriving Latitude from a Given Circumference **Description:** Determines the latitude corresponding to a known circumference, using the inverse of the adjusted formula. **Formula:** 𝑙=2𝜋⋅6378137⋅cos⁡(𝜙)⋅(1−0.00669438sin⁡2(𝜙))−0.5l=2π⋅6378137⋅cos(ϕ)⋅(1−0.00669438sin2(ϕ))−0.5 **Example Calculation:** Given 𝑙=36,788l=36,788 km, solve for 𝜙ϕ iteratively. ![[Attachments/1798894363.pdf]]