ky Miles]] [[Sky Mile Slide Deck]] [[The Sky Mile Presentation]] [[Sidereal Solar and Lunar Time]] [[Coordinate Systems]] [[Coordinate System and Map Projection Papers]] [[Longitude]] [[Maps and the Coordinate Systems they are Projected From]] [[Cosmography]] [[G Projector Map Software]] [[Attachments/Optics Derived .docx|Optics Derived ]] [[Optics Angular Resolution or Earth Curve]] [[Angular Resolution and our World]] [[Basic Trigonometry angular size, distance, and]] [[The Celestial Sphere]] [[69 Miles Per Degree]] Coordinate Conversions and Transformations including Formulas https://www.iogp.org/wp-content/uploads/2019/09/373-07-02.pdf https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates https://www.ngs.noaa.gov/PUBS_LIB/CoordinateCoversionforHydrographicSurveying_TR_NOS114_CGS7.pdf https://www.slideshare.net/JezreelDavid1/cylindrical-and-spherical-coordinates-system https://www.slideshare.net/suganthithangaraj1/coordinate-systems-57444755 Spherical coordinates https://mathinsight.org/spherical_coordinates Cartesian coordinates https://mathinsight.org/cartesian_coordinates Cylindrical coordinates https://mathinsight.org/cylindrical_coordinates Cylindrical and Spherical Coordinates https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates 8.2: Spherical and Cylindrical Coordinates https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/08%3A_Some_Curvilinear_Coordinate_Systems/8.02%3A_Spherical_and_Cylindrical_Coordinates ## **Coordinate Systems** [[Coordinate Systems]] [[Coordinate System and Map Projection Papers]] ![[Attachments/Pasted image 20240513022512.png]] ![[Attachments/Pasted image 20240512233440.png]] ![[Attachments/Pasted image 20240516233502.png]] ![[Attachments/Pasted image 20240516233511.png]] ![[Attachments/Pasted image 20240516233532.png]] ![[Attachments/Pasted image 20240516233541.png]] ![[Attachments/Pasted image 20240516233725.png]] ![[Attachments/Pasted image 20240516233738.png]] ![[Attachments/Pasted image 20240516233759.png]] ![[Attachments/Pasted image 20240516233807.png]] ![[Attachments/Pasted image 20240516233816.png]] ![[Attachments/Pasted image 20240516233829.png]] ![[Attachments/Pasted image 20240516233839.png]] ![[Attachments/Pasted image 20240516233846.png]] ![[Attachments/Pasted image 20240516233854.png]] ![[Attachments/Pasted image 20240516235540.png]] ![[Attachments/Pasted image 20240516235519.png]] Sure! Here are the relationships and conversion formulas between Cartesian coordinates (𝑥,𝑦,𝑧)(x,y,z), spherical coordinates (𝜌,𝜃,𝜙)(ρ,θ,ϕ), and cylindrical coordinates (𝑟,𝜃,𝑧)(r,θ,z) in a copy-and-paste-friendly format: ### 1. Cartesian to Spherical Coordinates - **Cartesian Coordinates**: (𝑥,𝑦,𝑧)(x,y,z) - **Spherical Coordinates**: (𝜌,𝜃,𝜙)(ρ,θ,ϕ) - 𝜌ρ (radial distance): Distance from the origin to the point. - 𝜃θ (azimuthal angle): Angle in the 𝑥𝑦xy-plane from the positive 𝑥x-axis. - 𝜙ϕ (polar angle): Angle from the positive 𝑧z-axis. ρ = sqrt(x^2 + y^2 + z^2) θ = atan2(y, x) φ = acos(z / ρ) ### 2. Spherical to Cartesian Coordinates - **Spherical Coordinates**: (𝜌,𝜃,𝜙)(ρ,θ,ϕ) - **Cartesian Coordinates**: (𝑥,𝑦,𝑧)(x,y,z) x = ρ * sin(φ) * cos(θ) y = ρ * sin(φ) * sin(θ) z = ρ * cos(φ) ### 3. Cartesian to Cylindrical Coordinates - **Cartesian Coordinates**: (𝑥,𝑦,𝑧)(x,y,z) - **Cylindrical Coordinates**: (𝑟,𝜃,𝑧)(r,θ,z) - 𝑟r (radial distance): Distance from the 𝑧z-axis to the point. - 𝜃θ (azimuthal angle): Angle in the 𝑥𝑦xy-plane from the positive 𝑥x-axis. - 𝑧z (height): Same as the 𝑧z-coordinate in Cartesian coordinates r = sqrt(x^2 + y^2) θ = atan2(y, x) z = z ### 4. Cylindrical to Cartesian Coordinates - **Cylindrical Coordinates**: (𝑟,𝜃,𝑧)(r,θ,z) - **Cartesian Coordinates**: (𝑥,𝑦,𝑧)(x,y,z) x = r * cos(θ) y = r * sin(θ) z = z ### 5. Spherical to Cylindrical Coordinates - **Spherical Coordinates**: (𝜌,𝜃,𝜙)(ρ,θ,ϕ) - **Cylindrical Coordinates**: (𝑟,𝜃,𝑧)(r,θ,z) r = ρ * sin(φ) θ = θ z = ρ * cos(φ) ### 6. Cylindrical to Spherical Coordinates - **Cylindrical Coordinates**: (𝑟,𝜃,𝑧)(r,θ,z) - **Spherical Coordinates**: (𝜌,𝜃,𝜙)(ρ,θ,ϕ) ρ = sqrt(r^2 + z^2) θ = θ φ = atan2(r, z) ![[Attachments/Pasted image 20240628123437.png]] ![[Attachments/Pasted image 20240513023243.png]] ![[Attachments/Pasted image 20240513003837.png]] Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal. The three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical. https://skytonight.org/coordinates 'Three Coordinate Systems Sphere of Stars - The night sky looks like an upside down [bowl](http://kerrymagruder.com/bcp/aster/general/bowl.htm) set on the horizon, but as it turns around during the night it is easy to think of it as a giant sphere. To think of the stars as lying on the inside surface of a giant **celestial sphere** which rotates around us once a day explains the appearances of [diurnal motion](http://kerrymagruder.com/bcp/diurnal/index.htm) with simplicity and elegance. With good reason this explanatory scheme was adopted by ancient Greek astronomers, beginning with the 6th century B.C. Pythagoreans, and it is remains the most convenient way to learn observational astronomy today. - Any **rotating sphere** has two poles at each end of the axis of rotation, and an equator which bisects the sphere in a plane that is perpendicular to the axis of rotation. Use a celestial globe model to identify the north and south **celestial poles** and the **celestial equator**. - Become familiar with a **model celestial globe** of the sort used in the planetarium labs. Note that the constellations depicted on these models appear reversed, since you're on the "outside looking in." Look through and across a model celestial sphere to inspect the constellations as they appear from the earth. - The celestial sphere concept facilitates the use of coordinate systems using imaginary lines inscribed on the celestial sphere. These lines rotate with the celestial sphere, and therefore do not depend on the observer's location, time of observation, or horizon. - **Purpose:** Any coordinate system provides a grid of perpendicular lines by which it is possible to specify the **unique location** of any single point on the celestial sphere. - **Base:** Any spherical coordinate system is based on a **great circle**, which bisects the celestial sphere into two equal hemispheres. - Example: Is the horizon a great circle? - **Step 1:** Each coordinate system begins with a measurement made **along** the great circle. - Example: A measurement along the horizon is called the **azimuth**. - **Step 2:** Each coordinate system also involves measurements **above or below** the great circle. - Example: A measurement above or below the horizon is called the **altitude**. - **How:** Measurements in angular degrees are made with a [quadrant](http://kerrymagruder.com/bcp/instruments/quadrant/index.htm). - A familiar coordinate system is based on the [Earth's equator](http://kerrymagruder.com/bcp/sphere/terreslat.htm) (terrestrial longitude and latitude). - Three great circles are used as the basis of three different celestial coordinate systems: - [Horizon](http://kerrymagruder.com/bcp/aster/general/bowl.htm) (altitude and azimuth) - [Celestial Equator](http://kerrymagruder.com/bcp/sphere/celequator.htm) (right ascension and declination) - [Ecliptic](http://kerrymagruder.com/bcp/sphere/ecliptic.htm) (celestial latitude and celestial longitude) ![[Attachments/Top-row-The-three-coordinate-systems-4-6-on-the-half-sphere-H-Bottom-row-The 1.png]] [[celestial sp]] ### Coordinate Systems Based on Celestial Observations ##### Geographic Coordinate Systems 1. **Geographic Coordinate System (Latitude and Longitude)** - **Based on Celestial Observations**: Yes. - **Details**: Latitude measurements are traditionally derived from observing the angle between the horizon and the North Star or the sun at noon. Longitude is determined by measuring the local time of a known position (like Greenwich) against the local solar time, which could historically only be done accurately at sea with the help of a chronometer. Before leaving the subject of specialized coordinate systems, we should say something about the coordinate systems that measure the surface of the earth. To an excellent approximation the shape of the earth is that of an oblate spheroid. This can cause some problems with the meaning of local vertical. **a. The Astronomical Coordinate System** The traditional coordinate system for locating positions on the surface of the earth is the latitude-longitude coordinate system. Most everyone has a feeling for this system as the latitude is simply the angular distance north or south of the equator measured along the local meridian toward the pole while the longitude is the angular distance measured along the equator to the local meridian from some reference meridian. This reference meridian has historically be taken to be that through a specific instrument (the Airy transit) located in Greenwich England. By a convention recently adopted by the International Astronomical Union, longitudes measured east of Greenwich are considered to be positive and those measured to the west are considered to be negative. Such coordinates provide a proper understanding for a perfectly spherical earth. But for an earth that is not exactly spherical, more care needs to be taken. **b. The Geodetic Coordinate System** In an attempt to allow for a non-spherical earth, a coordinate system has been devised that approximates the shape of the earth by an oblate spheroid. Such a figure can be generated by rotating an ellipse about its minor axis, which then forms the axis of the coordinate system. The plane swept out by the major axis of the ellipse is then its equator. This approximation to the actual shape of the earth is really quite good. The geodetic latitude is now given by the angle between the local vertical and the plane of the equator where the local vertical is the normal to the oblate spheroid at the point in question. The geodetic longitude is roughly the same as in the astronomical coordinate system and is the angle between the local meridian and the meridian at Greenwich. The difference between the local vertical (i.e. the normal to the local surface) and the astronomical vertical (defined by the local gravity vector) is known as the "deflection of the vertical" and is usually less than 20 arc-sec. The oblatness of the earth allows for the introduction of a third coordinate system sometimes called the geocentric coordinate system. **c. The Geocentric Coordinate System** Consider the oblate spheroid that best fits the actual figure of the earth. Now consider a radius vector from the center to an arbitrary point on the surface of that spheroid. In general, that radius vector will not be normal to the surface of the oblate spheroid (except at the poles and the equator) so that it will define a different local vertical. This in turn can be used to define a different latitude from either the astronomical or geodetic latitude. For the earth, the maximum difference between the geocentric and geodetic latitudes occurs at about 45° latitude and amounts to about (11' 33"). While this may not seem like much, it amounts to about eleven and a half nautical miles (13.3 miles or 21.4 km.) on the surface of the earth. Thus, if you really want to know where you are you must be careful which coordinate system you are using. Again the geocentric longitude is defined in the same manner as the geodetic longitude, namely it is the angle between the local meridian and the meridian at Greenwich. ##### Alt-Azimuth Coordinate System 1. **Horizontal Coordinate System (Altitude and Azimuth)** - **Based on Celestial Observations**: Yes. - **Details**: Used primarily in astronomy, this system measures the altitude of celestial bodies above the horizon and their azimuth, which is the angular distance measured along the horizon from the north. The Altitude-Azimuth coordinate system is the most familiar to the general public. The origin of this coordinate system is the observer and it is rarely shifted to any other point. The fundamental plane of the system contains the observer and the horizon. While the horizon is an intuitively obvious concept, a rigorous definition is needed as the apparent horizon is rarely coincident with the location of the true horizon. To define it, one must first define the zenith. This is the point directly over the observer's head, but is more carefully defined as the extension of the local gravity vector outward through the celestial sphere. This point is known as the astronomical zenith. Except for the oblatness of the earth, this zenith is usually close to the extension of the local radius vector from the center of the earth through the observer to the celestial sphere. The presence of large masses nearby (such as a mountain) could cause the local gravity vector to depart even further from the local radius vector. The horizon is then that line on the celestial sphere which is everywhere 90° from the zenith. The altitude of an object is the angular distance of an object above or below the horizon measured along a great circle passing through the object and the zenith. The azimuthal angle of this coordinate system is then just the azimuth of the object. The only problem here arises from the location of the zero point. Many older books on astronomy will tell you that the azimuth is measured westward from the south point of the horizon. However, only astronomers did this and most of them don't anymore. Surveyors, pilots and navigators, and virtually anyone concerned with local coordinate systems measures the azimuth from the north point of the horizon increasing through the east point around to the west. That is the position that I take throughout this book. Thus the azimuth of the cardinal points of the compass are: N(0°), E(90°), S(180°), W(270°). ##### The Right Ascension - Declination Coordinate System 1. **Equatorial Coordinate System (Right Ascension and Declination)** - **Based on Celestial Observations**: Yes. - **Details**: This system is used in astronomy to fix the position of stars and other celestial bodies. Right ascension is analogous to terrestrial longitude, and declination is analogous to latitude. It is based on the Earth's rotation axis and the celestial equator. This coordinate system is a spherical-polar coordinate system where the polar angle, instead of being measured from the axis of the coordinate system, is measured from the system's equatorial plane. Thus the declination is the angular complement of the polar angle. Simply put, it is the angular distance to the astronomical object measured north or south from the equator of the earth as projected out onto the celestial sphere. For measurements of distant objects made from the earth, the origin of the coordinate system can be taken to be at the center of the earth. At least the 'azimuthal' angle of the coordinate system is measured in the proper fashion. That is, if one points the thumb of his right hand toward the North Pole, then the fingers will point in the direction of increasing Right Ascension. Some remember it by noting that the Right Ascension of rising or ascending stars increases with time. There is a tendency for some to face south and think that the angle should increase to their right as if they were looking at a map. This is exactly the reverse of the true situation and the notion so confused air force navigators during the Second World War that the complementary angle, known as the sidereal hour angle, was invented. This angular coordinate is just 24 hours minus the Right Ascension. ##### **Geographic Coordinate System (Latitude and Longitude)** The Geographic Coordinate System (GCS) is a globally accepted system for defining locations on the Earth's surface using latitude and longitude. It inherently assumes a curved, spherical or ellipsoidal Earth, making it a three-dimensional coordinate system. Longitude and latitude form a coordinate system used to pinpoint locations on the Earth's surface. - **Latitude**: Measures north-south position between the poles and the equator. The equator represents 0 degrees latitude, while the poles are at 90 degrees north and south. Latitude lines, or parallels, run parallel to the equator. - **Longitude**: Measures east-west position and is expressed in degrees east or west from the Prime Meridian, which passes through Greenwich, England. Longitude lines, or meridians, converge at the poles and are widest at the equator. **Lat/Lon Invariance:** Both flat and spherical Earth models use latitude and longitude for navigation and mapping. The azimuthal transformation maintains these coordinates invariant, meaning they do not change between models. This is crucial for preserving distances calculated by common formulas like the Haversine, which calculates distances between two points on a sphere based on their latitudes and longitudes. ##### Right Ascension and Declination Right ascension (RA) and declination (Dec) are the celestial equivalents of longitude and latitude, respectively, used to specify the location of celestial objects. - **Declination**: Measures the angular distance of a point north or south of the celestial equator. It is analogous to latitude on Earth and is measured in degrees, minutes, and seconds. The celestial equator is 0 degrees declination, while the poles are at +90 degrees (north celestial pole) and -90 degrees (south celestial pole). - **Right Ascension**: Measures the angular distance of a celestial object eastward along the celestial equator from the vernal equinox. It is the celestial equivalent of longitude. However, right ascension is measured in hours, minutes, and seconds, where 24 hours complete a full circle (360 degrees), making each hour equivalent to 15 degrees. Certainly! Here are the main coordinate systems based on angles to the stars, commonly used in astronomy to pinpoint celestial objects: ### 1. **Horizontal Coordinate System (Altitude-Azimuth)** - **Origin**: The observer's location. - **Fundamental Plane**: The horizon. - **Coordinates**: - **Altitude (Alt)**: The angle between the object and the observer's horizon. It ranges from 0° (on the horizon) to 90° (at the zenith, directly overhead). - **Azimuth (Az)**: The angle measured clockwise from the north point along the horizon to the point directly below the celestial object. It ranges from 0° to 360°. - **Usage**: Common for amateur astronomy and for quick locating of celestial objects from an observer’s specific location. ### 2. **Equatorial Coordinate System (Right Ascension - Declination)** - **Origin**: The center of the Earth (geocentric) or the observer’s location (topocentric). - **Fundamental Plane**: The celestial equator, an extension of the Earth's equator into space. - **Coordinates**: - **Right Ascension (RA)**: Analogous to longitude, it measures the angular distance eastward along the celestial equator from the vernal equinox. It is measured in hours, minutes, and seconds, with 24 hours making a full circle. - **Declination (Dec)**: Analogous to latitude, it measures the angular distance north or south of the celestial equator. It ranges from +90° at the north celestial pole to -90° at the south celestial pole. - **Usage**: Standard for professional and amateur astronomy, used to catalog the positions of stars and other celestial objects. ### 3. **Ecliptic Coordinate System** - **Origin**: The center of the Earth or the observer’s location. - **Fundamental Plane**: The plane of the Earth’s orbit around the Sun (the ecliptic). - **Coordinates**: - **Ecliptic Longitude (λ)**: Measured along the ecliptic from the vernal equinox, ranging from 0° to 360°. - **Ecliptic Latitude (β)**: Measured perpendicular to the ecliptic, ranging from +90° to -90°. - **Usage**: Useful for tracking the positions of planets, the Moon, and the Sun relative to the Earth's orbit. ### 4. **Galactic Coordinate System** - **Origin**: The center of the Milky Way galaxy. - **Fundamental Plane**: The plane of the Milky Way galaxy. - **Coordinates**: - **Galactic Longitude (l)**: Measured along the plane of the Milky Way, starting from the direction of the Galactic Center, ranging from 0° to 360°. - **Galactic Latitude (b)**: Measured perpendicular to the galactic plane, ranging from +90° to -90°. - **Usage**: Primarily used in galactic astronomy to map the positions and motions of objects within the Milky Way. ### 5. **Supergalactic Coordinate System** - **Origin**: The center of the Local Supercluster. - **Fundamental Plane**: The plane of the Local Supercluster. - **Coordinates**: - **Supergalactic Longitude (SGL)**: Measured along the plane of the Local Supercluster. - **Supergalactic Latitude (SGB)**: Measured perpendicular to the supergalactic plane. - **Usage**: Used in large-scale structure studies of the universe, mapping the distribution of galaxies and galaxy clusters. ### Summary of Coordinate Systems Based on Angles to the Stars: 1. **Horizontal Coordinate System (Altitude-Azimuth)**: - **Coordinates**: Altitude, Azimuth - **Usage**: Local observer-specific, simple and intuitive for casual observation. 2. **Equatorial Coordinate System (Right Ascension - Declination)**: - **Coordinates**: Right Ascension, Declination - **Usage**: Standard astronomical coordinate system for star catalogs and precise observations. 3. **Ecliptic Coordinate System**: - **Coordinates**: Ecliptic Longitude, Ecliptic Latitude - **Usage**: Useful for solar system objects, aligning with the plane of Earth's orbit. 4. **Galactic Coordinate System**: - **Coordinates**: Galactic Longitude, Galactic Latitude - **Usage**: Mapping the Milky Way, studying the structure and contents of our galaxy. 5. **Supergalactic Coordinate System**: - **Coordinates**: Supergalactic Longitude, Supergalactic Latitude - **Usage**: Studying the large-scale structure of the universe, mapping galaxy distributions. These coordinate systems allow astronomers to precisely locate and study celestial objects in relation to the Earth, the solar system, and the broader universe. ## the curvature of the celestial sphere Coordinate systems based on angles to the stars account for the curvature of the celestial sphere by treating the sky as a spherical surface surrounding the observer. This allows for the precise measurement of positions and movements of celestial objects relative to the observer. Here's how these systems manage the curvature of the celestial sphere: ### 1. Concept of the Celestial Sphere - **Celestial Sphere**: An imaginary sphere of arbitrarily large radius centered on the Earth. All celestial objects (stars, planets, etc.) are assumed to lie on this sphere for the purpose of mapping their positions. - **Observer's Perspective**: The observer is at the center of this sphere, with the sky appearing to be a dome above them. ### 2. Coordinate Systems and the Celestial Sphere #### Horizontal Coordinate System (Altitude-Azimuth) - **Origin**: The observer's location. - **Curvature**: Takes into account the apparent curvature of the sky as seen from the observer's vantage point. - **Coordinates**: - **Altitude (Alt)**: The angle between the object and the observer's horizon, ranging from 0° at the horizon to 90° at the zenith. - **Azimuth (Az)**: The angle measured clockwise from the north point along the horizon to the point directly below the celestial object, ranging from 0° to 360°. **Curvature Handling**: - The altitude measures how high an object is in the sky, directly accounting for the observer’s horizon. - The azimuth provides the horizontal direction, with the horizon forming the fundamental plane. #### Equatorial Coordinate System (Right Ascension - Declination) - **Origin**: The center of the Earth (geocentric) or the observer’s location (topocentric). - **Curvature**: Accounts for the Earth's rotation and the celestial sphere's curvature. - **Coordinates**: - **Right Ascension (RA)**: Analogous to longitude, measured eastward along the celestial equator from the vernal equinox. - **Declination (Dec)**: Analogous to latitude, measured north or south of the celestial equator. **Curvature Handling**: - The celestial equator is the projection of the Earth’s equator onto the celestial sphere. - Right ascension and declination define positions on this curved surface, similar to latitude and longitude on Earth. #### Ecliptic Coordinate System - **Origin**: The center of the Earth. - **Curvature**: Based on the plane of Earth's orbit around the Sun (the ecliptic plane). - **Coordinates**: - **Ecliptic Longitude (λ)**: Measured along the ecliptic from the vernal equinox. - **Ecliptic Latitude (β)**: Measured perpendicular to the ecliptic plane. **Curvature Handling**: - The ecliptic plane is inclined at about 23.5° to the celestial equator. - This system takes into account the curvature of the celestial sphere relative to the ecliptic plane. #### Galactic Coordinate System - **Origin**: The center of the Milky Way galaxy. - **Curvature**: Uses the plane of the Milky Way as the fundamental plane. - **Coordinates**: - **Galactic Longitude (l)**: Measured along the plane of the Milky Way. - **Galactic Latitude (b)**: Measured perpendicular to the galactic plane. **Curvature Handling**: - Aligns with the structure of the Milky Way, projecting this structure onto the celestial sphere. ### 3. Practical Implementation To implement these coordinate systems, several steps are involved: - **Defining the Fundamental Plane**: Each system defines a fundamental plane (horizon, celestial equator, ecliptic, or galactic plane) that helps in positioning objects on the curved celestial sphere. - **Measurement of Angles**: Angles are measured relative to the fundamental plane and other reference points (e.g., vernal equinox for right ascension, north point for azimuth). - **Spherical Trigonometry**: Calculations on the celestial sphere use spherical trigonometry to account for the curvature. This allows for accurate transformations between different coordinate systems. - **Transformation Equations**: Equations transform coordinates between different systems (e.g., from horizontal to equatorial coordinates). ### Example: Transformation from Horizontal to Equatorial Coordinates - **Inputs**: Altitude (Alt), Azimuth (Az), observer's latitude (φ). - **Outputs**: Right Ascension (RA), Declination (Dec). - **Process**: 1. Calculate the hour angle (HA) from the azimuth and altitude. 2. Use the hour angle and observer’s latitude to determine the declination. 3. Convert the hour angle to right ascension using the local sidereal time. ### Conclusion Coordinate systems based on angles to the stars handle the curvature of the celestial sphere by using spherical geometry principles and defining appropriate reference planes. This ensures accurate measurement and representation of celestial positions, making these systems essential tools in both amateur and professional astronomy. ##### Polar Coordinate System (Radial Distance, Polar Angle) ![[Attachments/aimimg_2-3183395903.jpg]] ![[Attachments/sneha-c-polar-coordinates-06-1605002801-2205063269.png]] ![[Attachments/1200pxPoint_in_Polar_coordinates_gridsvg-3853665859.png]] ![[Attachments/Pasted image 20240513005455.png]] Details**: Used in mathematics and fields requiring circular or rotational measurements, this system is based on the distance from a central point and the angle from a reference direction (usually the positive x-axis). Coordinate Curves: 𝑟r-curves (Circles): These are circles of constant radius 𝑅R, centered at the origin. In navigation, these can represent lines of equal distance from a central point, such as ranges from a specific sensor or landmark. 𝜃θ-curves (Radial Lines): These lines emanate from the origin at a constant angle 𝜃θ. They are akin to bearings or azimuths used in navigation to define a direction from a specific point. Transformation Specifics:** The transformation from polar to Cartesian coordinates (and vice versa) is crucial for tasks such as plotting positions on a map or defining movement trajectories in navigation, where directions and distances from a central reference are common. Non-uniqueness and Mapping Considerations: The text mentions that the polar coordinate transformation is not one-to-one in general due to the periodic nature of the angle 𝜃θ. However, by restricting 𝜃θ to a single full rotation (0, 2π)) and 𝑟r to positive values, the transformation becomes one-to-one onto the 𝑥𝑦xy-plane, omitting the origin. This restriction is important in navigation and mapping to avoid ambiguity in position representation. #### Interconnection and Applications 1. **Geometric Parallels**: - Both systems are based on great circles: latitude and declination are measured from the equatorial planes (Earth’s equator and celestial equator), while longitude and right ascension are measured from prime meridians (Greenwich for Earth and the vernal equinox for the sky). - Each uses a fundamental circle (equator) and a starting point for measuring east-west coordinates (Greenwich meridian for longitude, vernal equinox for right ascension). 1. **Navigational Uses**: - Navigators have historically used the positions of celestial objects along these coordinates to determine their location on Earth. For instance, by measuring the altitude of Polaris (the North Star), one can determine their latitude in the Northern Hemisphere, since its declination is nearly +90 degrees. - Longitude on Earth was historically more challenging to determine until the advent of accurate timekeeping, which allowed for the comparison of local solar time against a reference time at a known longitude. In celestial terms, this is akin to measuring the hour angle of a celestial body to determine right ascension. 1. **Time and Angular Measurement**: - Time is intrinsically linked to these coordinate systems. Earth’s rotation, which defines the measurement of a day, affects both longitude (through time zones) and right ascension (through the sidereal day, about four minutes shorter than the solar day). - Angular measurements on Earth and the celestial sphere facilitate calculations of distance and time. For example, as Earth rotates at about 15 degrees per hour, timekeeping is crucial for determining longitude and for astronomical observations using right ascension. - 1. **Celestial Navigation** - The practice of celestial navigation involves measuring the angles between celestial bodies and the horizon, and comparing these observations with tables based on right ascension and declination. This data can then be translated into terrestrial latitude and longitude for navigation. The systems of longitude and latitude and right ascension and declination are fundamental to understanding our place in the universe and navigating within it. These systems underscore the universality of spherical geometry in describing both our planet and the stars. #### Connecting Coordinate Transformation with Measurement: 1. **Equivalence in Different Models:** - When you perform calculations such as distances using azimuthal transformations, whether the Earth is considered flat or spherical in model, the distances like those measured in nautical miles remain consistent. This is because the formulas that calculate distances based on angles (like the Haversine formula) remain valid regardless of the underlying shape assumption of the Earth. ![[Attachments/Pasted image 20240513001425.png]] When you perform calculations such as distances using azimuthal transformations, whether the Earth is considered flat or spherical in model, the distances like those measured in nautical miles remain consistent. **This is because the formulas that calculate distances based on angles (like the Haversine formula) remain valid regardless of the underlying shape assumption of the Earth.** Practical Implication This theoretical equivalence suggests that for navigation and mapping, using either a flat or spherical model does not impact the practical outcomes when using properly adjusted coordinate transformations. Measurements like nautical and statute miles retain their values and utility because their definition ties back to the coordinate system (latitude and longitude), which remains invariant between transformations. ### Coordinate Systems Not Based on Celestial Observations ##### **Cartesian Coordinate System (X, Y, Z) **Details**: A purely mathematical system used in various fields including mathematics, physics, engineering, and everyday life. Coordinates are based on perpendicular axes originating from a defined origin point. In the cartesian coordinate system, there are **three** coordinate axes, each at a right angle with each other. These axes are name **_x_**, **_y_**, and **_z_**. In a right-handed cartesian system, the thumb points x-axis, the forefinger points y-axis, and the middle finger points z-axis. A point in the cartesian coordinate system is represented by 3 values along these axes. These values relate to the length of the vector along that particular axis from the origin. The cartesian coordinate system is shown in Fig. 2. We describe the other related terms needed in vector analysis inside the cartesian coordinate system, but these terms are applicable to vectors in any coordinate system. These terms are necessary to proceed to the next coordinate systems. The surfaces have an area of dxdy, ##### UTM (Universal Transverse Mercator) and MGRS (Military Grid Reference System) -Details**: These systems divide the Earth into zones and use metric measurements on a flat plane. They are based on a specific map projection (transverse Mercator), which is not directly derived from celestial observations but from mathematical formulae for the projection. ##### Polar Coordinate System (Radial Distance, Polar Angle) ![[Attachments/aimimg_2-3183395903.jpg]] ![[Attachments/sneha-c-polar-coordinates-06-1605002801-2205063269.png]] ![[Attachments/1200pxPoint_in_Polar_coordinates_gridsvg-3853665859.png]] ![[Attachments/Pasted image 20240513005455.png]] Details**: Used in mathematics and fields requiring circular or rotational measurements, this system is based on the distance from a central point and the angle from a reference direction (usually the positive x-axis). Coordinate Curves: 𝑟r-curves (Circles): These are circles of constant radius 𝑅R, centered at the origin. In navigation, these can represent lines of equal distance from a central point, such as ranges from a specific sensor or landmark. 𝜃θ-curves (Radial Lines): These lines emanate from the origin at a constant angle 𝜃θ. They are akin to bearings or azimuths used in navigation to define a direction from a specific point. Transformation Specifics:** The transformation from polar to Cartesian coordinates (and vice versa) is crucial for tasks such as plotting positions on a map or defining movement trajectories in navigation, where directions and distances from a central reference are common. Non-uniqueness and Mapping Considerations: The text mentions that the polar coordinate transformation is not one-to-one in general due to the periodic nature of the angle 𝜃θ. However, by restricting 𝜃θ to a single full rotation (0, 2π)) and 𝑟r to positive values, the transformation becomes one-to-one onto the 𝑥𝑦xy-plane, omitting the origin. This restriction is important in navigation and mapping to avoid ambiguity in position representation. ![[Attachments/Pasted image 20240513005300.png]] ![[Attachments/Pasted image 20240513022528.png]] ![[Attachments/Pasted image 20240513005324.png]] ![[Attachments/Pasted image 20240513005335.png]] ### Geographic Coordinate System Longitude Latitude Relation to Earth's Spherical Geometry Why Not a Circle? According to Consensus and GPT: When considering a circle in two dimensions, the concept of latitude and longitude does not apply because there is no curvature to account for. A circle lacks the three-dimensional aspect of a sphere that causes distances and directions to change based on position relative to the Earth's axis. While you could measure the circumference of a circle and divide it into units, this measurement would not provide the necessary information to navigate the three-dimensional surface of the Earth, where direction and distance are influenced by spherical geometry. Therefore, while the **mathematical** principle of dividing a circle into degrees and minutes could theoretically apply to any circle, the specific application and utility of a nautical mile are inherently linked to navigating across the curved surface of the Earth. The spherical geometry is what allows navigators to use the concept of nautical miles to calculate distances and plot courses accurately, considering the Earth's curvature. HA But the truth is: - **Spherical Earth:** A nautical mile is defined as one minute (1/60th of a degree) of latitude. Given that there are 360 degrees in a sphere and each degree consists of 60 minutes, the Earth's circumference along a meridian is about 21,600 nautical miles. When you consider the Earth as a flat circle (like in some flat Earth models or specific map projections such as the azimuthal equidistant projection, which the Gleason map utilizes), the nautical mile could theoretically still be defined in relation to the circumference of this circle: - **Circular Model:** Here, if you take the circle's full circumference to analogously represent a 360-degree rotation around the center, you can maintain the same division into degrees and minutes. Thus, in this model, a nautical mile (as 1/60th of a degree) could still be used as a unit of measure based on the circle's circumference. - **Latitude and Longitude:** The Earth's surface is divided into lines of latitude and longitude, which form a grid system used for navigation. Latitudes are parallel lines that circle the Earth, measuring the distance north or south of the equator. Longitudes, however, converge at the poles and measure the distance east or west of the Prime Meridian. The definition of a nautical mile is based on minutes of latitude, which directly correspond to the Earth's spherical surface. One minute of arc along any meridian (lines running from pole to pole) is uniformly equal to one nautical mile, irrespective of the observer’s location on the Earth. This consistency stems from the spherical shape of the Earth, where the meridians are great circles. What these really are, are fundamentally based on measurements derived from observing celestial bodies and are designed for a spherical model of the Earth. ![[Attachments/3 (2) 1.png]] ![[Attachments/2 (2) 2.png]] ![[Attachments/4 (2) 1.png]] The concepts of longitude and latitude are fundamentally based on measurements derived from observing celestial bodies and are designed for a spherical model of the Earth. These geographic coordinates form a grid system used to pinpoint locations on the globe. Here’s a detailed breakdown: 1. **Latitude**: - **Celestial Observations**: Latitude is determined by measuring the angle between the horizon and the North Star (Polaris) in the Northern Hemisphere. The angle is equivalent to the latitude of the observer. In the Southern Hemisphere, similar measurements are made with other celestial bodies like the Southern Cross. - **Relation to the Earth’s Geometry**: Latitude lines are parallel to the equator and indicate a location’s distance north or south of the equator, which is defined as 0° latitude. These lines are imaginary circles on the surface of the Earth, with the size of each circle decreasing as one moves toward the poles. 1. **Longitude**: - **Celestial Observations**: Longitude determination historically relied on the accurate measurement of time and the observation of the positions of celestial bodies. The invention of the marine chronometer by John Harrison in the 18th century allowed navigators to measure the time difference between a fixed location (like Greenwich, England) and their current position at sea, enabling them to calculate how far east or west they had traveled. - **Relation to the Earth’s Geometry**: Lines of longitude, or meridians, run perpendicular to lines of latitude and converge at the poles. They measure the distance east or west of the Prime Meridian (0° longitude), which passes through Greenwich. Latitude and Longitude Based on a Globe latitude and longitude are indeed based on observations of the sky (celestial navigation) and are fundamentally tied to a spherical model of the Earth. The entire system of latitude and longitude is based on the premise that the Earth is spherical. This is evident in several ways: - **Great Circles**: Both latitude and longitude lines are based on the concept of great circles that divide the globe into equal halves. The equator and all meridians are great circles. - **Spherical Trigonometry**: The calculations for distances and angles between different points on the Earth’s surface use spherical trigonometry, assuming the Earth is a sphere. - **Navigation and Mapping**: All navigational and mapping systems that use latitude and longitude take into account the Earth’s curvature. For example, flight routes and maritime courses plotted using these coordinates reflect adjustments for the globe’s shape. - **Great Circles vs. Circles:** A great circle is the largest circle that can be drawn on a sphere's surface, dividing it into two equal halves. On a sphere like the Earth, the shortest distance between two points lies along the arc of a great circle. - providing the most efficient and shortest path between two points. - In contrast, a circle (in the context of a two-dimensional plane) does not account for the varying distances and directions encountered on a three-dimensional spherical surface due to curvature ,A great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.[1 ![[Attachments/Pasted image 20240513063408.png]]][2] Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non-antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the minor arc, and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius. Any other circle of the sphere is called a small circle, and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1. [[Attachments/Pasted image 20240509112429.jpeg|Open: Pasted image 20240509112429.png]] ![[Attachments/Pasted image 20240509112429.jpeg]] ## Types of Coordinate Systems relative to us #### Spherical ![[Attachments/Pasted image 20240513023831.png]] ![[Attachments/Pasted image 20240513023840.png]] ![[Attachments/Pasted image 20240513024544.png]] ![[Attachments/Pasted image 20240513023331.png]] ![[Attachments/Pasted image 20240513023340.png]] ![[Attachments/Pasted image 20240513023359.png]] ![[Attachments/Pasted image 20240513023415.png]] ![[Attachments/Pasted image 20240513025019.png]] ![[Attachments/Pasted image 20240513024558.png]] ![[Attachments/Pasted image 20240513024606.png]] ![[Attachments/Pasted image 20240513024617.png]] ![[Attachments/Pasted image 20240513024626.png]] ![[Attachments/Pasted image 20240513024636.png]] ![[Attachments/Pasted image 20240513024645.png]] ![[Attachments/Pasted image 20240513024655.png]] ![[Attachments/Pasted image 20240513023856.png]] ![[Attachments/Pasted image 20240513023904.png]] ![[Attachments/Pasted image 20240513023914.png]] ![[Attachments/Pasted image 20240513023929.png]] #### Cylindrical https://www.slideshare.net/nisargamin6236/cylindrical-co-ordinate-system-63715673 ![[Attachments/Pasted image 20240513023538.png]] ![[Attachments/Pasted image 20240513023547.png]] ![[Attachments/Pasted image 20240513023602.png]] ![[Attachments/Pasted image 20240513023609.png]] ![[Attachments/Pasted image 20240513023619.png]] ![[Attachments/Pasted image 20240513023809.png]] ![[Attachments/Pasted image 20240513023819.png]] ![[Attachments/Pasted image 20240513024524.png]] ![[Attachments/Pasted image 20240513024458.png]] ![[Attachments/Pasted image 20240513024505.png]] ![[Attachments/Pasted image 20240513024513.png]] https://www.slideshare.net/suganthithangaraj1/coordinate-systems-57444755 #### Cylindrical and Spherical ![[Attachments/Pasted image 20240513023642.png]] https://www.slideshare.net/JezreelDavid1/cylindrical-and-spherical-coordinates-system ![[Attachments/maxresdefault (4).jpg]] ![[Attachments/Pasted image 20240513023720.png]] ![[Attachments/Pasted image 20240513023728.png]] ![[Attachments/Pasted image 20240513023735.png]] ![[Attachments/Pasted image 20240513023744.png]] ![[Attachments/Pasted image 20240513024915.png]] ![[Attachments/Pasted image 20240513024923.png]] ![[Attachments/Pasted image 20240513025053.png]] ![[Attachments/Pasted image 20240513025105.png]] ![[Attachments/Pasted image 20240513025113.png]] Azimuthal transformation is a coordinate transformation that converts cartesian space into a cylinder. The transformation is lossless and reversible ![[Attachments/Pasted image 20240513024729.png]] ![[Attachments/Pasted image 20240513025004.png]] ![[Attachments/Pasted image 20240513024844.png]] ### Cartesian and Spherical ![[Attachments/Cylindrical-and-spherical-coordinates-2-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-3-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-4-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-6-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-9-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-13-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-14-2048 1.jpg]] ### Spherical to Rectangular ![[Attachments/Cylindrical-and-spherical-coordinates-21-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-17-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-18-2048.jpg]] ### Spherical to Rectangular ![[Attachments/Cylindrical-and-spherical-coordinates-19-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-20-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-25-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-26-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-27-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-28-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-29-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-30-2048.jpg]] ![[Attachments/Cylindrical-and-spherical-coordinates-31-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-9-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-10-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-11-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-12-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-13-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-14-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-15-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-16-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-17-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-18-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-18-2048 1.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-19-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-20-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-21-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-22-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-23-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-24-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-26-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-27-2048.jpg]] ![[Attachments/Cylindrical-and-Spherical-Coordinates-System-28-2048.jpg]] ### Deriving the circumfrence To deduce the circumference of Earth from the observation of a single arc minute, you can follow a simple calculation based on the historical definition of a nautical mile. Historically, one nautical mile was defined as one arc minute of latitude along any meridian. Given that an arc minute is 1/60th of a degree, and there are 360 degrees in a full circle, you can calculate the circumference of Earth as follows: 1. **Understand the Relationship:** One arc minute corresponds to one nautical mile. There are 60 minutes in one degree and 360 degrees in a full circle. 2. **Calculate Total Arc Minutes Around Earth:** Since there are 360 degrees in a circle and each degree contains 60 arc minutes, the total number of arc minutes around the Earth is 360×60=21,360×60=21,600 arc minutes. 3. **Deduce Earth's Circumference:** If each arc minute corresponds to one nautical mile, then the Earth's circumference is 21,600 nautical miles. Using the modern definition of a nautical mile as exactly 1,852 meters, the calculation for Earth's circumference in meters would be: 21,600 nautical miles×1,852 meters/nautical mile=39,931,200 meters21,600nautical miles×1,852meters/nautical mile=39,931,200meters So, from the observation of a single arc minute, you could deduce that the circumference of Earth is approximately 39,931.2 kilometers (or about 40,075 kilometers, considering the actual average circumference of the Earth). This method is a simplification and assumes a perfectly spherical Earth for the sake of the calculation. However, this provides a close approximation and demonstrates the fundamental concept used historically to define the nautical mile and deduce Earth's size. ![[Attachments/COORDINATE-SYSTEM-pdf-3-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-4-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-5-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-6-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-7-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-8-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-9-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-10-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-11-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-12-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-13-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-14-2048.jpg]] ![[Attachments/COORDINATE-SYSTEM-pdf-15-2048.jpg]] Coordinate Conversions and Transformations including Formulas https://www.iogp.org/wp-content/uploads/2019/09/373-07-02.pdf https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates https://www.ngs.noaa.gov/PUBS_LIB/CoordinateCoversionforHydrographicSurveying_TR_NOS114_CGS7.pdf Coordinate conversion·is used in a general sense in this report to denote the process of· changing the coordi nates of a position represented in one reference system to the coordi nates of that position represented in any other reference system. Coordi nate conversions invol ve one or more coordi nate transformations. . Coordi nate transformation appl ies specifical ly to the process of converti ng geodetic coordinates to pl anar coordi nates based on the same elli psoid (forward transformation ), converti ng planar coordi nates to geodetic coordi nates based on the same el l i psoid (inverse transformation ), and converti ng geodetic coordinates to geodetic coordinates based on a different el l i psoid (datum transformation). Transformations are basic operations in performing the more general coordi nate conversions. A mappi ng projecti on. or simply projection , is a system whereby geodeti c coordinates and planar coordi nates are related with a one-to-one correspondence. (Note that the tenn "projection" appl ies here to mappings between mathemati cal surfaces, not to the projection of a poi nt on the physi cal surface of the Earth to.a mathemati cal surface .) Mapping projections are defi ned by specifyi ng certain conditions that must be met . When a projecti on is referred by name, those condi tions are impl ied. For exampl e. in the transverse Mercator projection. angl es between infinitesimal l i ne segments are preserved, and scale is held constant al ong a selected meridian . Mappi ng projections are further defined by specifyi ng certain projection parameters, thereby orienti ng the pl ane or devel opable surface to the el l i psoid. 11.1 Introduction to Cartesian Coordinates in Space https://sites.und.edu/timothy.prescott/apex/web/apex.Ch11.S1.html Cyclindrical https://sites.und.edu/timothy.prescott/apex/web/figures/figspacecylinder1b_3D.html Sections https://sites.und.edu/timothy.prescott/apex/web/figures/figspacecylinder1_3D.html A coordinate system is a set of mathematical rules for specifying how coordinates are to be assigned to points. It includes the definition of the coordinate axes, the units to be used and the geometry of the axes. A coordinate system is an abstract concept, unrelated to the Earth. A coordinate system is related to the Earth through a datum. The combination of coordinate system and datum is a coordinate reference system (CRS). ## Coordinate Operations Coordinates may be changed from one coordinate reference system to another through the application of a coordinate operation. Two types of coordinate operation may be distinguished: • coordinate conversion, where no change of datum is involved and the parameters are chosen and thus error free. • coordinate transformation, where the target CRS is based on a different datum to the source CRS. ![[Attachments/Pasted image 20240512221358.png]] Transformation parameters are empirically determined and thus subject to measurement errors.** ** The coordinate systems of astronomical importance are nearly all spherical coordinate systems. The obvious reason for this is that most all astronomical objects are remote from the earth and so appear to move on the backdrop of the celestial sphere. While one may still use a spherical coordinate system for nearby objects, it may be necessary to choose the origin to be the observer to avoid problems with parallax. These orthogonal coordinate frames will differ only in the location of the origin and their relative orientation to one another. Since they have their foundation in observations made from the earth, their relative orientation is related to the orientation of the earth's rotation axis with respect to the stars and the sun. The most important of these coordinate systems is the Right Ascension -Declination coordinate system. https://ads.harvard.edu/books/1989fcm..book/Chapter2.pdf?bcsi-ac-1890e3206a556864=2791AF9A000000027+JSzzQ3B4QhD6Sh3wQCtNjc1S5NEQAAAgAAAMn7QACEAwAAGwAAAC7oCwA= ## orthogonal and Nonorthogonal coordinate systems An orthogonal system is one in which the coordinates arc mutually perpendicular. Nonorthogonal systems are hard to work with and they are of little or no practical use. Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal. ![[Attachments/Pasted image 20240516233457.png]] 1 A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordi nate system may turn out to be easy in another system. In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical. https://x-lumin.com/wp-content/uploads/2020/09/Coordinate_Transforms.pdf Earth Centered Earth Fixed (ECEF) Coordinates ECEF coordinates are dened by the World Geodetic System 1984 (WGS84) standard. Note; this is a geodetic coordinate system not a geocentric coordinate system. The dierence between geodetic and geocentric is: the down direction in a geodetic coordinate system does not point to the center of the Earth, rather it points normal to the Earth's surface, whereas in a geocentric down always points to the center of the earth. This because the Earth is an oblate spheroid, not a sphere. See the red r direction in Figure 1. ![[Attachments/Pasted image 20240512232106.png]] Moving your data between coordinate systems sometimes includes transforming between the geographic coordinate systems. Because the geographic coordinate systems contain datums that are based on spheroids, a geographic transformation also changes the underlying spheroid. There are several methods, which have different levels of accuracy and ranges, for transforming between datums. The accuracy of a particular transformation can range from centimeters to meters depending on the method and the quality and number of control points available to define the transformation parameters. A geographic transformation always converts geographic (longitude–latitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates. ![[Attachments/Pasted image 20240513022123.png]] ![[Attachments/Pasted image 20240513022134.png]] Three-parameter methods The simplest datum transformation method is a geocentric, or three-parameter, transformation. The geocentric transformation models the differences between two datums in the X,Y,Z coordinate system. One datum is defined with its center at 0,0,0. The center of the other datum is defined at some distance (∆X,∆Y,∆Z) in meters away Usually the transformation parameters are defined as going ‘from’ a local datum ‘to’ WGS 1984 or another geocentric datum. The three parameters are linear shifts and are always in meters. Seven-parameter methods A more complex and accurate datum transformation is possible by adding four more parameters to a geocentric transformation. The seven parameters are three linear shifts (∆X,∆Y,∆Z), three angular rotations around each axis (rx ,ry ,rz ), and scale factor(s). ( ) X Y Z X Y Z s r r r r r r X Y Z new z y z x y x original The rotation values are given in decimal seconds, while the scale factor is in parts per million (ppm). The rotation values are defined in two different ways. It’s possible to define the rotation angles as positive either clockwise or counterclockwise as you look toward the origin of the X,Y,Z systems. The equation in the previous column is how the United States and Australia define the equations and is called the Coordinate Frame Rotation transformation. The rotations are positive counterclockwise. Europe uses a different convention called the Position Vector transformation. Both methods are sometimes referred to as the Bursa–Wolf method. In the Projection Engine, the Coordinate Frame and Bursa–Wolf methods are the same. Both Coordinate Frame and Position Vector methods are supported, and it is easy to convert transformation values from one method to the other simply by changing the signs of the three rotation values. For example, the parameters to convert from the WGS 1972 datum to the WGS 1984 datum with the Coordinate Frame method are (in the order, ∆X, ∆Y,∆Z,rx ,ry ,rz ,s): 
 To use the same parameters with the Position Vector method, change the sign of the rotation so the new parameters are:  ![[Attachments/Pasted image 20240513022235.png]] ![[Attachments/Pasted image 20240513022245.png]] ## Projections vs Transformations VS Translations ![[Attachments/Pasted image 20240512221627.png]] ![[Attachments/Pasted image 20240512221637.png]] ![[Attachments/Pasted image 20240512221646.png]] They are all just the Graticule projected. ![[Attachments/Pasted image 20240512221740.png]] # Types of Coordinate Systems 101 ## Coordinate Systems Based on Celestial Observations #### Geographic Coordinate Systems 1. **Geographic Coordinate System (Latitude and Longitude)** - **Based on Celestial Observations**: Yes. - **Details**: Latitude measurements are traditionally derived from observing the angle between the horizon and the North Star or the sun at noon. Longitude is determined by measuring the local time of a known position (like Greenwich) against the local solar time, which could historically only be done accurately at sea with the help of a chronometer. Before leaving the subject of specialized coordinate systems, we should say something about the coordinate systems that measure the surface of the earth. To an excellent approximation the shape of the earth is that of an oblate spheroid. This can cause some problems with the meaning of local vertical. **a. The Astronomical Coordinate System** The traditional coordinate system for locating positions on the surface of the earth is the latitude-longitude coordinate system. Most everyone has a feeling for this system as the latitude is simply the angular distance north or south of the equator measured along the local meridian toward the pole while the longitude is the angular distance measured along the equator to the local meridian from some reference meridian. This reference meridian has historically be taken to be that through a specific instrument (the Airy transit) located in Greenwich England. By a convention recently adopted by the International Astronomical Union, longitudes measured east of Greenwich are considered to be positive and those measured to the west are considered to be negative. Such coordinates provide a proper understanding for a perfectly spherical earth. But for an earth that is not exactly spherical, more care needs to be taken. **b. The Geodetic Coordinate System** In an attempt to allow for a non-spherical earth, a coordinate system has been devised that approximates the shape of the earth by an oblate spheroid. Such a figure can be generated by rotating an ellipse about its minor axis, which then forms the axis of the coordinate system. The plane swept out by the major axis of the ellipse is then its equator. This approximation to the actual shape of the earth is really quite good. The geodetic latitude is now given by the angle between the local vertical and the plane of the equator where the local vertical is the normal to the oblate spheroid at the point in question. The geodetic longitude is roughly the same as in the astronomical coordinate system and is the angle between the local meridian and the meridian at Greenwich. The difference between the local vertical (i.e. the normal to the local surface) and the astronomical vertical (defined by the local gravity vector) is known as the "deflection of the vertical" and is usually less than 20 arc-sec. The oblatness of the earth allows for the introduction of a third coordinate system sometimes called the geocentric coordinate system. **c. The Geocentric Coordinate System** Consider the oblate spheroid that best fits the actual figure of the earth. Now consider a radius vector from the center to an arbitrary point on the surface of that spheroid. In general, that radius vector will not be normal to the surface of the oblate spheroid (except at the poles and the equator) so that it will define a different local vertical. This in turn can be used to define a different latitude from either the astronomical or geodetic latitude. For the earth, the maximum difference between the geocentric and geodetic latitudes occurs at about 45° latitude and amounts to about (11' 33"). While this may not seem like much, it amounts to about eleven and a half nautical miles (13.3 miles or 21.4 km.) on the surface of the earth. Thus, if you really want to know where you are you must be careful which coordinate system you are using. Again the geocentric longitude is defined in the same manner as the geodetic longitude, namely it is the angle between the local meridian and the meridian at Greenwich. ### Alt-Azimuth Coordinate System 1. **Horizontal Coordinate System (Altitude and Azimuth)** - **Based on Celestial Observations**: Yes. - **Details**: Used primarily in astronomy, this system measures the altitude of celestial bodies above the horizon and their azimuth, which is the angular distance measured along the horizon from the north. The Altitude-Azimuth coordinate system is the most familiar to the general public. The origin of this coordinate system is the observer and it is rarely shifted to any other point. The fundamental plane of the system contains the observer and the horizon. While the horizon is an intuitively obvious concept, a rigorous definition is needed as the apparent horizon is rarely coincident with the location of the true horizon. To define it, one must first define the zenith. This is the point directly over the observer's head, but is more carefully defined as the extension of the local gravity vector outward through the celestial sphere. This point is known as the astronomical zenith. Except for the oblatness of the earth, this zenith is usually close to the extension of the local radius vector from the center of the earth through the observer to the celestial sphere. The presence of large masses nearby (such as a mountain) could cause the local gravity vector to depart even further from the local radius vector. The horizon is then that line on the celestial sphere which is everywhere 90° from the zenith. The altitude of an object is the angular distance of an object above or below the horizon measured along a great circle passing through the object and the zenith. The azimuthal angle of this coordinate system is then just the azimuth of the object. The only problem here arises from the location of the zero point. Many older books on astronomy will tell you that the azimuth is measured westward from the south point of the horizon. However, only astronomers did this and most of them don't anymore. Surveyors, pilots and navigators, and virtually anyone concerned with local coordinate systems measures the azimuth from the north point of the horizon increasing through the east point around to the west. That is the position that I take throughout this book. Thus the azimuth of the cardinal points of the compass are: N(0°), E(90°), S(180°), W(270°). ### The Right Ascension - Declination Coordinate System 1. **Equatorial Coordinate System (Right Ascension and Declination)** - **Based on Celestial Observations**: Yes. - **Details**: This system is used in astronomy to fix the position of stars and other celestial bodies. Right ascension is analogous to terrestrial longitude, and declination is analogous to latitude. It is based on the Earth's rotation axis and the celestial equator. This coordinate system is a spherical-polar coordinate system where the polar angle, instead of being measured from the axis of the coordinate system, is measured from the system's equatorial plane. Thus the declination is the angular complement of the polar angle. Simply put, it is the angular distance to the astronomical object measured north or south from the equator of the earth as projected out onto the celestial sphere. For measurements of distant objects made from the earth, the origin of the coordinate system can be taken to be at the center of the earth. At least the 'azimuthal' angle of the coordinate system is measured in the proper fashion. That is, if one points the thumb of his right hand toward the North Pole, then the fingers will point in the direction of increasing Right Ascension. Some remember it by noting that the Right Ascension of rising or ascending stars increases with time. There is a tendency for some to face south and think that the angle should increase to their right as if they were looking at a map. This is exactly the reverse of the true situation and the notion so confused air force navigators during the Second World War that the complementary angle, known as the sidereal hour angle, was invented. This angular coordinate is just 24 hours minus the Right Ascension. ## Coordinate Systems Not Based on Celestial Observations 1. **Cartesian Coordinate System (X, Y, Z)** - **Based on Celestial Observations**: No. - **Details**: A purely mathematical system used in various fields including mathematics, physics, engineering, and everyday life. Coordinates are based on perpendicular axes originating from a defined origin point. Cartesian coordinate system In the cartesian coordinate system, there are **three** coordinate axes, each at a right angle with each other. These axes are name **_x_**, **_y_**, and **_z_**. In a right-handed cartesian system, the thumb points x-axis, the forefinger points y-axis, and the middle finger points z-axis. A point in the cartesian coordinate system is represented by 3 values along these axes. These values relate to the length of the vector along that particular axis from the origin. The cartesian coordinate system is shown in Fig. 2. We describe the other related terms needed in vector analysis inside the cartesian coordinate system, but these terms are applicable to vectors in any coordinate system. These terms are necessary to proceed to the next coordinate systems. The surfaces have an area of dxdy, ## Polar Coordinates **Polar coordinates**, system of locating points in a plane with reference to a fixed point O (the origin) and a ray from the origin usually chosen to be the positive x-axis. The [coordinates](https://www.britannica.com/science/coordinate-system) are written (_r,__θ_), in which _r_is the distance from the origin to any desired point P and _θ_is the angle made by the line OP and the axis. A simple relationship exists between [Cartesian coordinates](https://www.britannica.com/science/Cartesian-coordinates)(_x,y_) and the polar coordinates (_r,__θ_)_,_namely: _x_= _r_cos _θ,_and _y_= _r_sin _θ_. An [analog](https://www.merriam-webster.com/dictionary/analog) of polar coordinates, called spherical coordinates, may also be used to locate points in three-dimensional space. The system used involves again the distance from the origin O to a given point P, the angle _θ,_measured between OP and the positive _z_axis, and a second angle _ϕ,_measured between the positive _x_axis and the projection of OP onto the _x,y_plane. Those angles are essentially the colatitude and longitude used to express locations on the Earth’s surface, where the colatitude is 90 degrees minus the latitude. - **Definition**: Polar coordinates define a point in a plane using a distance from a reference point (radius 𝑟) and an angle (𝜃) from a reference direction (usually the positive x-axis of a Cartesian coordinate system). - **Use Cases**: Commonly used in mathematics and fields involving circular or rotational symmetry, such as physics (e.g., analyzing movements in circular paths), engineering, and computer graphics. - **Characteristics**: - The origin is called the pole. - The radius (r) indicates how far from the pole the point is. - The angle (𝜃) indicates the direction from the reference line (often the horizontal axis). Defining Polar Coordinates To find the coordinates of a point in the polar coordinate system, consider [Figure 1.27](https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates#CNX_Calc_Figure_11_03_001). The point P𝑃 has Cartesian coordinates (x,y).(𝑥,𝑦). The line segment connecting the origin to the point P𝑃 measures the distance from the origin to P𝑃 and has length r.𝑟. The angle between the positive x𝑥-axis and the line segment has measure θ.𝜃. This observation suggests a natural correspondence between the coordinate pair (x,y)(𝑥,𝑦) and the values r𝑟 and θ.𝜃. This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term _ordered pair_) associated with it. In the polar coordinate system, each point also has two values associated with it: r𝑟 and θ. ![[Attachments/Pasted image 20240513013642.png]] ![[Attachments/Pasted image 20240513013650.png]] Each point (x,y) in the Cartesian coordinate system can therefore be represented as an ordered pair (r,θ) in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate. Every point in the plane can be represented in this form. Note that the equation tanθ=y/xhas an infinite number of solutions for any ordered pair (x,y).(𝑥,𝑦). However, if we restrict the solutions to values between 00 and 2π2𝜋 then we can assign a unique solution to the quadrant in which the original point (𝑥,𝑦) is located. Then the corresponding value of _r_ is positive, so r2=x2+y2. ![[Attachments/Pasted image 20240513013718.png]] ![[Attachments/Pasted image 20240513013745.png]] The line segment starting from the center of the graph going to the right (called the positive _x_-axis in the Cartesian system) is the polar axis. The center point is the pole, or origin, of the coordinate system, and corresponds to r=0.𝑟=0. The innermost circle shown in [Figure 1.28](https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates#CNX_Calc_Figure_11_03_002) contains all points a distance of 1 unit from the pole, and is represented by the equation r=1.𝑟=1. Then r=2𝑟=2 is the set of points 2 units from the pole, and so on. The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of r𝑟 is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle. We have now seen several examples of drawing graphs of curves defined by polar equations. A summary of some common curves is given in the tables below. In each equation, _a_ and _b_ are arbitrary constants.  Figure 1.31  Figure 1.32 A cardioid is a special case of a limaçon (pronounced “lee-mah-son”), in which a=b𝑎=𝑏 or a=−b.𝑎=−𝑏. The rose is a very interesting curve. Notice that the graph of r=3sin2θ𝑟=3sin2𝜃 has four petals. However, the graph of r=3sin3θ𝑟=3sin3𝜃 has three petals as shown.  Figure 1.33 Graph of r=3sin3θ.𝑟=3sin3𝜃. If the coefficient of θ𝜃 is even, the graph has twice as many petals as the coefficient. If the coefficient of θ𝜃 is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of θ𝜃 is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started ([Figure 1.34](https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates#CNX_Calc_Figure_11_03_009)(a)). However, if the coefficient is irrational, then the curve never closes ([Figure 1.34](https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates#CNX_Calc_Figure_11_03_009)(b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive _x_ axis are slightly thicker. This is because the petal does not quite match up with the starting point.  Figure 1.34 Polar rose graphs of functions with (a) rational coefficient and (b) irrational coefficient. Note that the rose in part (b) does not fill the disk of radius 3 centered at the origin, even though it appears to. To see this, just note that for any fixed ray from the origin, there are only a countable number of values of theta for which we get a point on the graph along that ray. What we can say, though, is that for any point in the disk, there are points as close as you please to that point on the given graph. 1. **UTM (Universal Transverse Mercator) and MGRS (Military Grid Reference System)** - **Based on Celestial Observations**: No. - **Details**: These systems divide the Earth into zones and use metric measurements on a flat plane. They are based on a specific map projection (transverse Mercator), which is not directly derived from celestial observations but from mathematical formulae for the projection. 1. **Polar Coordinate System (Radial Distance, Polar Angle)** - **Based on Celestial Observations**: No. - **Details**: Used in mathematics and fields requiring circular or rotational measurements, this system is based on the distance from a central point and the angle from a reference direction (usually the positive x-axis). ## Hybrid Systems - **ECEF (Earth-Centered, Earth-Fixed)** - **Based on Celestial Observations**: Partially. - **Details**: This system uses a 3D Cartesian coordinate model where the origin is the center of mass of the Earth. While not based on celestial angles, it is designed to align with the Earth’s rotation axis, which relates to celestial mechanics. https://faculty.umb.edu/michael.trust/EnvSci360_S17_Lecture3.pdf https://faculty.umb.edu/michael.trust/EnvSci360_S17_Lecture3.pdf ![[Attachments/Pasted image 20240513005521.png]] ## Cartesian coordinate system In the cartesian coordinate system, there are **three** coordinate axes, each at a right angle with each other. These axes are name **_x_**, **_y_**, and **_z_**. In a right-handed cartesian system, the thumb points x-axis, the forefinger points y-axis, and the middle finger points z-axis. A point in the cartesian coordinate system is represented by 3 values along these axes. These values relate to the length of the vector along that particular axis from the origin. The cartesian coordinate system is shown in Fig. 2. We describe the other related terms needed in vector analysis inside the cartesian coordinate system, but these terms are applicable to vectors in any coordinate system. These terms are necessary to proceed to the next coordinate systems. The surfaces have an area of dxdy, ![[Attachments/Pasted image 20240512233408.png]] https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates ### Conversion between Cylindrical and Cartesian ![[Attachments/Pasted image 20240513005607.png]] ![[Attachments/Pasted image 20240513005634.png]] ![[Attachments/Pasted image 20240513005637.png]] ![[Attachments/Pasted image 20240513005656.png]] ![[Attachments/Pasted image 20240513005711.png]] ## ## Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances (randz)(𝑟and𝑧) and an angle measure (θ).(𝜃). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates. In the spherical coordinate system, a point P𝑃 in space ([Figure 2.97](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_011)) is represented by the ordered triple (ρ,θ,φ)(𝜌,𝜃,𝜑) where - $𝜌$ (the Greek letter rho) is the distance between P𝑃 and the origin (ρ≠0);(𝜌≠0); - $θ$ is the same angle used to describe the location in cylindrical coordinates; - $φ$ (the Greek letter phi) is the angle formed by the positive _z_-axis and line segment  - $OP––––,$𝑂𝑃—, where $O$ is the origin and $0≤φ≤π.$ ![[Attachments/Pasted image 20240513005754.png]] https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation $x2+y2+z2=c2𝑥2+𝑦2+𝑧2=𝑐2$ has the simple equation $𝜌=𝑐$ in spherical coordinates. In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in. Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 4000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.  Figure 2.104 In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian. Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive _z_-axis. The prime meridian represents the trace of the surface as it intersects the _xz_-plane. The equator is the trace of the sphere intersecting the _xy_-plane. ### Converting from Spherical Coordinates Plot the point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) and express its location in both rectangular and cylindrical coordinates. ### Solution Use the equations in [Converting among Spherical, Cylindrical, and Rectangular Coordinates](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#fs-id1163723844895) to translate between spherical and cylindrical coordinates ([Figure 2.100](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_014)): $x=ρsinφcosθ=8sin(π6)cos(π3)=8(12)12=2$ $y=ρsinφsinθ=8sin(π6)sin(π3)=8(12)3√2=23–√z=ρcosφ=8cos(π6)=8(3√2)=43–√.𝑥=𝜌sin𝜑cos𝜃=8sin(𝜋6)cos(𝜋3)=8(12)12=2$ $𝑦=𝜌sin𝜑sin𝜃=8sin(𝜋6)sin(𝜋3)=8(12)32=23𝑧=𝜌cos𝜑=8cos(𝜋6)=8(32)=43$ ![[Attachments/Pasted image 20240513013046.png]]  Figure 2.100 The projection of the point in the _xy_-plane is 44 units from the origin. The line from the origin to the point’s projection forms an angle of π/3𝜋/3 with the positive _x_-axis. The point lies 43–√43 units above the _xy_-plane. The point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) has rectangular coordinates (2,23–√,43–√).(2,23,43). Finding the values in cylindrical coordinates is equally straightforward: $rθz===ρsinφ=8sinπ6=4θρcosφ=8cosπ6=43–√.𝑟=𝜌sin𝜑=8sin𝜋6=4𝜃=𝜃𝑧=𝜌cos𝜑=8cos𝜋6=43.$ Thus, cylindrical coordinates for the point are (4,π3,43–√).(4,𝜋3,43). ![[Attachments/Pasted image 20240513013118.png]] Spherical Coordinates: https://www.geogebra.org/m/xnyhgvzz ![[Attachments/Pasted image 20240513070622.png]] https://www.geogebra.org/m/tspwn49p ![[Attachments/Pasted image 20240513070734.png]] ### . Cartesian System: - **Coordinate Surfaces**: X=constant, Y=constant, Z=constant. - **Description**: In this system, each constant coordinate surface is a plane. These planes are orthogonal to each other: - X=constant forms a plane parallel to the YZ plane. - Y=constant forms a plane parallel to the XZ plane. - Z=constant forms a plane parallel to the XY plane. ### 2. Cylindrical System: - **Coordinate Surfaces**: ρ=constant, Φ=constant, Z=constant. - **Description**: - **ρ=constant** results in a circular cylinder centered on the Z-axis. - **Φ=constant** results in a semi-infinite plane that extends radially from the Z-axis and is orthogonal to it. - **Z=constant** produces a horizontal plane that cuts across the cylindrical structure, similar to the Z=constant in the Cartesian system. ### 3. Spherical System: - **Coordinate Surfaces**: r=constant, θ=constant, Φ=constant. - **Description**: - **r=constant** forms a sphere centered at the origin. - **θ=constant** results in a circular cone with the vertex at the origin and symmetric about the Z-axis. - **Φ=constant** is a semi-infinite plane that divides the space into two, similar to longitude lines on a globe. ### Relationship and Comparison: - **Orthogonality**: In each system, the surfaces corresponding to constant values of one of the coordinates are orthogonal to the direction of the coordinate axis associated with the varying coordinate. For instance, in the Cartesian system, planes orthogonal to the X-axis are formed by Y=constant and Z=constant. - **Shape Variance**: While the Cartesian system uses only planes, the cylindrical system introduces cylindrical surfaces, and the spherical system uses spherical and conical surfaces. This reflects how these systems adapt to different applications – Cartesian for general purposes, cylindrical for situations with rotational symmetry (like around a central axis), and spherical for global representations (like Earth). - **Utility**: Each system is particularly useful depending on the symmetry and needs of the problem at hand. Cartesian is used for general 3D space, cylindrical is used for systems with an axis of rotation (such as around pipes or circular towers), and spherical is best for large-scale phenomena centered around a point (like gravitational fields around planets or geographic coordinates on Earth). https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.7%3A_Cylindrical_and_Spherical_Coordinates # Spherical and Cylindrical Coordinates https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/08%3A_Some_Curvilinear_Coordinate_Systems/8.02%3A_Spherical_and_Cylindrical_Coordinates ![[Attachments/Pasted image 20240514232629.png]] The relationship between Cartesian coordinates (x,y,z)(𝑥,𝑦,𝑧) and cylindrical coordinates (r,θ,z)(𝑟,𝜃,𝑧) is given by x=cos(θ) Y=rsin(θ) Z=z ![[Attachments/Pasted image 20240513070137.png]] Consider now **spherical coordinates**, the second generalization of polar form in three dimensions. For a point (x,y,z)(𝑥,𝑦,𝑧) in three dimensional space, the spherical coordinates are defined as follows. ρ:the length of the ray from the origin to the pointθ:the angle between the positive x-axis and the ray from the origin to the point (x,y,0)ϕ:the angle between the positive z-axis and the ray from the origin to the point of interest𝜌:the length of the ray from the origin to the point𝜃:the angle between the positive 𝑥-axis and the ray from the origin to the point (𝑥,𝑦,0)𝜙:the angle between the positive 𝑧-axis and the ray from the origin to the point of interest The spherical coordinates are determined by (ρ,ϕ,θ)(𝜌,𝜙,𝜃). The relation between these and the Cartesian coordinates (x,y,z)(𝑥,𝑦,𝑧) for a point are as follows. xyz=ρsin(ϕ)cos(θ), ϕ∈[0,π]=ρsin(ϕ)sin(θ), θ∈[0,2π)=ρcosϕ, ρ≥0. ![[Attachments/Pasted image 20240513070200.png]] ![[Attachments/Pasted image 20240514232623.png]] Consider the pictures below. The first illustrates the surface when 𝜌 is known, which is a sphere of radius ρ𝜌. The second picture corresponds to knowing both 𝜌 and ϕ which results in a circle about the 𝑧-axis. Suppose the first picture demonstrates a graph of the Earth. Then the circle in the second picture would correspond to a particular latitude. ![[Attachments/Pasted image 20240513070223.png]] Giving the third coordinate, θ𝜃 completely specifies the point of interest. This is demonstrated in the following picture. If the latitude corresponds to ϕ𝜙, then we can think of θ𝜃 as the longitude. ![[Attachments/Pasted image 20240513070239.png]] ![[Attachments/Pasted image 20240513070248.png]] Therefore, we can represent the same point in three ways, using Cartesian coordinates, (x,y,z)(𝑥,𝑦,𝑧), cylindrical coordinates, (r,θ,z)(𝑟,𝜃,𝑧), and spherical coordinates (ρ,ϕ,θ)(𝜌,𝜙,𝜃). Using this picture to review, call the point of interest P𝑃 for convenience. The Cartesian coordinates for P𝑃 are (x,y,z)(𝑥,𝑦,𝑧). Then ρ𝜌 is the distance between the origin and the point P𝑃. The angle between the positive z𝑧 axis and the line between the origin and P𝑃 is denoted by ϕ𝜙. Then θ𝜃 is the angle between the positive x𝑥 axis and the line joining the origin to the point (x,y,0)(𝑥,𝑦,0) as shown. This gives the spherical coordinates, (ρ,ϕ,θ)(𝜌,𝜙,𝜃). Given the line from the origin to (x,y,0)(𝑥,𝑦,0), r=ρsin(ϕ)𝑟=𝜌sin⁡(𝜙) is the length of this line. Thus r𝑟 and θ𝜃 determine a point in the xy𝑥𝑦-plane. In other words, r𝑟 and θ𝜃 are the usual polar coordinates and r≥0𝑟≥0 and θ∈[0,2π)𝜃∈[0,2𝜋). Letting z𝑧 denote the usual z𝑧 coordinate of a point in three dimensions, (r,θ,z)(𝑟,𝜃,𝑧) are the cylindrical coordinates of P𝑃. The relation between spherical and cylindrical coordinates is that r=ρsin(ϕ)𝑟=𝜌sin⁡(𝜙) and the θ𝜃 is the same as the θ𝜃 of cylindrical and polar coordinates. ![[Attachments/Pasted image 20240513070303.png]] ![[Attachments/Pasted image 20240514232759.png]] **Explanation:** 1. **Cartesian Coordinates (x, y, z)**: - Describes the position of point P in a traditional three-axis (three-dimensional) system. 2. **Cylindrical Coordinates (r, θ, z)**: - Converts the point's position into cylindrical format where: - r is the radial distance from the z-axis. - 𝜃 is the angular component measured from the positive x-axis in the xy-plane. - 𝑧 is the height along the z-axis, unchanged from Cartesian. 3. **Spherical Coordinates (ρ, φ, θ)**: - Converts the point's position into spherical format where: - 𝜌 is the radial distance from the origin to the point. - 𝜙 is the polar angle measured from the positive z-axis. - 𝜃 is the azimuthal angle, also measured in the xy-plane from the positive x-axis. The description also notes the specific transformations: - The radial distance r in cylindrical coordinates is derived from the spherical coordinates as 𝑟=𝜌sin⁡(𝜙 - The angular coordinate 𝜃θ remains consistent between the cylindrical and spherical systems. ## how to determine the position of a celestial object in the sky using astronomical coordinates, specifically explaining how the local hour angle (LHA), azimuth, and zenith distance are calculated. This process involves the celestial sphere and involves understanding how celestial navigation and observational astronomy work. Let’s break it down step-by-step for clarity. ### Understanding Key Concepts and Calculations #### 1. **Celestial Sphere:** - The celestial sphere is an imaginary sphere where the Earth is at the center. Celestial objects like stars, planets, and the Moon are projected onto this sphere for easier observation and measurement. #### 2. **Local Hour Angle (LHA):** - **Definition**: The Local Hour Angle is the angle measured westward along the celestial equator from the observer's meridian to the hour circle passing through the object. - **Calculation**: LHA can be calculated as follows: 𝐿𝐻𝐴=Local Sidereal Time (LST)−Right Ascension (RA)LHA=Local Sidereal Time (LST)−Right Ascension (RA) - LHA is crucial as it helps determine when a celestial object will be visible above the horizon at the observer's location. #### 3. **Right Ascension and Declination:** - These are coordinates on the celestial sphere akin to longitude and latitude on Earth. Right Ascension (RA) is equivalent to celestial longitude, and Declination (Dec) is similar to latitude. #### 4. **Sidereal Time:** - Sidereal Time is timekeeping based on Earth's rotation relative to the fixed stars (not the Sun). - **Local Sidereal Time (LST)** is determined from the observer’s longitude and the current date and time. #### 5. **Finding Local Azimuth and Zenith Distance:** - Once you have the LHA, you can use it with the declination of the object and the latitude of the observer to find the object's position in the sky. - **Azimuth (A)**: The direction (from north) along the horizon to the object. - **Zenith Distance (z)**: The angular distance from directly overhead (zenith) to the object. It is complementary to the altitude (altitude = 90° - z). #### 6. **Parallactic Angle (η):** - This angle is used for precise observations, particularly in astrophotography and photometric measurements. - **Purpose**: It helps in making corrections for atmospheric refraction, which affects the apparent position of a celestial object. ### Practical Application Here’s how these calculations are practically applied in observational astronomy or celestial navigation: 1. **Determine LST**: Calculate or look up the Local Sidereal Time based on your geographic longitude and a standard time reference. 2. **Calculate LHA**: Use LST and the Right Ascension of the celestial object to find the Local Hour Angle. 3. **Use Spherical Trigonometry**: With LHA, declination, and your latitude, you can solve the spherical triangle formed by the celestial poles, your zenith, and the object to find the azimuth and zenith distance. 4. **Adjust for Refraction**: Use the parallactic angle to adjust the observed position for atmospheric refraction, ensuring more accurate positioning or photometric data ## CHECKPOINT 2.58 Plot the point with spherical coordinates (2,−5π6,π6)(2,−5𝜋6,𝜋6) and describe its location in both rectangular and cylindrical coordinates. ## EXAMPLE 2.64 ### Converting from Rectangular Coordinates Convert the rectangular coordinates (−1,1,6–√)(−1,1,6) to both spherical and cylindrical coordinates. ## EXAMPLE 2.65 ### Identifying Surfaces in the Spherical Coordinate System Describe the surfaces with the given spherical equations. 1. θ=π3𝜃=𝜋3 2. φ=5π6𝜑=5𝜋6 3. ρ=6𝜌=6 4. ρ=sinθsinφ𝜌=sin𝜃sin𝜑 ### Solution 1. The variable θ𝜃 represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates (ρ,π3,φ)(𝜌,𝜋3,𝜑) lie on the plane that forms angle θ=π3𝜃=𝜋3 with the positive _x_-axis. Because ρ>0,𝜌>0, the surface described by equation θ=π3𝜃=𝜋3 is the half-plane shown in [Figure 2.101](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_016).  Figure 2.101 The surface described by equation θ=π3𝜃=𝜋3 is a half-plane. 2. Equation φ=5π6𝜑=5𝜋6 describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring 5π65𝜋6 rad with the positive _z_-axis. These points form a half-cone ([Figure 2.102](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_017)). Because there is only one value for φ𝜑 that is measured from the positive _z_-axis, we do not get the full cone (with two pieces).  Figure 2.102 The equation φ=5π6𝜑=5𝜋6 describes a cone. To find the equation in rectangular coordinates, use equation φ=arccos(zx2+y2+z2√).𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2). 5π6cos5π6−3√2343x24+3y24+3z243x24+3y24−z24======arccos(zx2+y2+z2√)zx2+y2+z2√zx2+y2+z2√z2x2+y2+z2z20.5𝜋6=arccos(𝑧𝑥2+𝑦2+𝑧2)cos5𝜋6=𝑧𝑥2+𝑦2+𝑧2−32=𝑧𝑥2+𝑦2+𝑧234=𝑧2𝑥2+𝑦2+𝑧23𝑥24+3𝑦24+3𝑧24=𝑧23𝑥24+3𝑦24−𝑧24=0. This is the equation of a cone centered on the _z_-axis. 3. Equation ρ=6𝜌=6 describes the set of all points 66 units away from the origin—a sphere with radius 66 ([Figure 2.103](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_018)).  Figure 2.103 Equation ρ=6𝜌=6 describes a sphere with radius 6.6. 4. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations y=ρsinφsinθ𝑦=𝜌sin𝜑sin𝜃 and ρ2=x2+y2+z2:𝜌2=𝑥2+𝑦2+𝑧2: ρρ2x2+y2+z2x2+y2−y+z2x2+y2−y+14+z2x2+(y−12)2+z2======sinθsinφρsinθsinφy01414.Multiply both sides of the equation byρ.Substitute rectangular variables using the equations above.Subtractyfrom both sides of the equation.Complete the square.Rewrite the middle terms as a perfect square.𝜌=sin𝜃sin𝜑𝜌2=𝜌sin𝜃sin𝜑Multiply both sides of the equation by𝜌.𝑥2+𝑦2+𝑧2=𝑦Substitute rectangular variables using the equations above.𝑥2+𝑦2−𝑦+𝑧2=0Subtract𝑦from both sides of the equation.𝑥2+𝑦2−𝑦+14+𝑧2=14Complete the square.𝑥2+(𝑦−12)2+𝑧2=14.Rewrite the middle terms as a perfect square. The equation describes a sphere centered at point (0,12,0)(0,12,0) with radius 12.12. ## CHECKPOINT 2.59 Describe the surfaces defined by the following equations. 1. ρ=13𝜌=13 2. θ=2π3𝜃=2𝜋3 3. φ=π4𝜑=𝜋4 Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x2+y2+z2=c2𝑥2+𝑦2+𝑧2=𝑐2 has the simple equation ρ=c𝜌=𝑐 in spherical coordinates. In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in [Figure 2.104](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_019). Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 40004000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.  Figure 2.104 In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian. Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive _z_-axis. The prime meridian represents the trace of the surface as it intersects the _xz_-plane. The equator is the trace of the sphere intersecting the _xy_-plane. ## EXAMPLE 2.66 ### Converting Latitude and Longitude to Spherical Coordinates The latitude of Columbus, Ohio, is 40°40° N and the longitude is 83°83° W, which means that Columbus is 40°40° north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40°.40°. In the same way, measuring from the prime meridian, Columbus lies 83°83° to the west. Express the location of Columbus in spherical coordinates. ### Solution The radius of Earth is 40004000 mi, so ρ=4000.𝜌=4000. The intersection of the prime meridian and the equator lies on the positive _x_-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83°,𝜃=−83°, Because Columbus lies 40°40° north of the equator, it lies 50°50° south of the North Pole, so φ=50°.𝜑=50°. In spherical coordinates, Columbus lies at point (4000,−83°,50°).(4000,−83°,50°). ## CHECKPOINT 2.60 Sydney, Australia is at 34°S34°S and 151°E.151°E. Express Sydney’s location in spherical coordinates. Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one. ## EXAMPLE 2.67 ### Choosing the Best Coordinate System In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. _Note_: There is not enough information to set up or solve these problems; we simply select the coordinate system ([Figure 2.105](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_021)). 1. Find the center of gravity of a bowling ball. 2. Determine the velocity of a submarine subjected to an ocean current. 3. Calculate the pressure in a conical water tank. 4. Find the volume of oil flowing through a pipeline. 5. Determine the amount of leather required to make a football.  Figure 2.105 (credit: (a) modification of work by scl hua, Wikimedia, (b) modification of work by DVIDSHUB, Flickr, (c) modification of work by Michael Malak, Wikimedia, (d) modification of work by Sean Mack, Wikimedia, (e) modification of work by Elvert Barnes, Flickr) ## CHECKPOINT 2.61 Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)? How should we orient the coordinate axes? ## Section 2.7 Exercises Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates (r,θ,z)(𝑟,𝜃,𝑧) of a point are given. Find the rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) of the point. [363](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723560107-solution). (4,π6,3)(4,𝜋6,3) 364. (3,π3,5)(3,𝜋3,5) [365](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723330516-solution). (4,7π6,3)(4,7𝜋6,3) 366. (2,π,−4)(2,𝜋,−4) For the following exercises, the rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) of a point are given. Find the cylindrical coordinates (r,θ,z)(𝑟,𝜃,𝑧) of the point. [367](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723561014-solution). (1,3–√,2)(1,3,2) 368. (1,1,5)(1,1,5) [369](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723186992-solution). (3,−3,7)(3,−3,7) 370. (−22–√,22–√,4)(−22,22,4) For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. [371](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723187112-solution). **[T]** r=4𝑟=4 372. **[T]** z=r2cos2θ𝑧=𝑟2cos2𝜃 [373](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723248454-solution). **[T]** r2cos(2θ)+z2+1=0𝑟2cos(2𝜃)+𝑧2+1=0 374. **[T]** r=3sinθ𝑟=3sin𝜃 [375](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723103383-solution). **[T]** r=2cosθ𝑟=2cos𝜃 376. **[T]** r2+z2=5𝑟2+𝑧2=5 [377](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723571904-solution). **[T]** r=2secθ𝑟=2sec𝜃 378. **[T]** r=3cscθ𝑟=3csc𝜃 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. [379](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724081932-solution). z=3𝑧=3 380. x=6𝑥=6 [381](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724081999-solution). x2+y2+z2=9𝑥2+𝑦2+𝑧2=9 382. y=2x2𝑦=2𝑥2 [383](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724019018-solution). x2+y2−16x=0𝑥2+𝑦2−16𝑥=0 384. x2+y2−3x2+y2−−−−−−√+2=0𝑥2+𝑦2−3𝑥2+𝑦2+2=0 For the following exercises, the spherical coordinates (ρ,θ,φ)(𝜌,𝜃,𝜑) of a point are given. Find the rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) of the point. [385](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724074697-solution). (3,0,π)(3,0,𝜋) 386. (1,π6,π6)(1,𝜋6,𝜋6) [387](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724074822-solution). (12,−π4,π4)(12,−𝜋4,𝜋4) 388. (3,π4,π6)(3,𝜋4,𝜋6) For the following exercises, the rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) of a point are given. Find the spherical coordinates (ρ,θ,φ)(𝜌,𝜃,𝜑) of the point. Express the measure of the angles in degrees rounded to the nearest integer. [389](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723572648-solution). (4,0,0)(4,0,0) 390. (−1,2,1)(−1,2,1) [391](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723183788-solution). (0,3,0)(0,3,0) 392. (−2,23–√,4)(−2,23,4) For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. [393](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723573545-solution). **[T]** ρ=3𝜌=3 394. **[T]** φ=π3𝜑=𝜋3 [395](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724049525-solution). **[T]** ρ=2cosφ𝜌=2cos𝜑 396. **[T]** ρ=4cscφ𝜌=4csc𝜑 [397](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724049712-solution). **[T]** φ=π2𝜑=𝜋2 398. **[T]** ρ=6cscφsecθ𝜌=6csc𝜑sec𝜃 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface. [399](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724066058-solution). x2+y2−3z2=0,𝑥2+𝑦2−3𝑧2=0, z≠0𝑧≠0 400. x2+y2+z2−4z=0𝑥2+𝑦2+𝑧2−4𝑧=0 [401](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723566851-solution). z=6𝑧=6 402. x2+y2=9𝑥2+𝑦2=9 For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φ𝜑 in radians rounded to four decimal places. [403](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723566962-solution). **[T]** (1,π4,3)(1,𝜋4,3) 404. **[T]** (5,π,12)(5,𝜋,12) [405](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724050254-solution). (3,π2,3)(3,𝜋2,3) 406. (3,−π6,3)(3,−𝜋6,3) For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. [407](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724050393-solution). (2,−π4,π2)(2,−𝜋4,𝜋2) 408. (4,π4,π6)(4,𝜋4,𝜋6) [409](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723299013-solution). (8,π3,π2)(8,𝜋3,𝜋2) 410. (9,−π6,π3)(9,−𝜋6,𝜋3) For the following exercises, find the most suitable system of coordinates to describe the solids. [411](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724057692-solution). The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length a,𝑎, where a>0𝑎>0 412. A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of a𝑎 and b,𝑏, respectively, where b>a>0𝑏>𝑎>0 [413](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723073046-solution). A solid inside sphere x2+y2+z2=9𝑥2+𝑦2+𝑧2=9 and outside cylinder (x−32)2+y2=94(𝑥−32)2+𝑦2=94 414. A cylindrical shell of height 1010 determined by the region between two cylinders with the same center, parallel rulings, and radii of 22 and 5,5, respectively [415](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723314891-solution). **[T]** Use a CAS to graph the region between elliptic paraboloid z=x2+y2𝑧=𝑥2+𝑦2 and cone x2+y2−z2=0.𝑥2+𝑦2−𝑧2=0. Then describe the region in cylindrical coordinates. 416. **[T]** Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the _xy_-plane between sphere x2+y2+z2=4𝑥2+𝑦2+𝑧2=4 and elliptical cone x2+y2−z2=0.𝑥2+𝑦2−𝑧2=0. [417](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723339550-solution). Washington, DC, is located at 39°39° N and 77°77° W (see the following figure). Assume the radius of Earth is 40004000 mi. Express the location of Washington, DC, in spherical coordinates. 418. San Francisco is located at 37.78°N37.78°N and 122.42°W.122.42°W. Assume the radius of Earth is 40004000 mi. Express the location of San Francisco in spherical coordinates. [419](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163724138611-solution). Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are (4000,−43.17°,102.91°).(4000,−43.17°,102.91°). 420. Find the latitude and longitude of Berlin if its spherical coordinates are (4000,13.38°,37.48°).(4000,13.38°,37.48°). [421](https://openstax.org/books/calculus-volume-3/pages/chapter-2#fs-id1163723341675-solution). **[T]** Consider the torus of equation (x2+y2+z2+R2−r2)2=4R2(x2+y2),(𝑥2+𝑦2+𝑧2+𝑅2−𝑟2)2=4𝑅2(𝑥2+𝑦2), where R≥r>0.𝑅≥𝑟>0. 1. Write the equation of the torus in spherical coordinates. 2. If R=r,𝑅=𝑟, the surface is called a _horn torus_. Show that the equation of a horn torus in spherical coordinates is ρ=2Rsinφ.𝜌=2𝑅sin𝜑. 3. Use a CAS to graph the horn torus with R=r=2𝑅=𝑟=2 in spherical coordinates. 422. **[T]** The “bumpy sphere” with an equation in spherical coordinates is ρ=a+bcos(mθ)sin(nφ),𝜌=𝑎+𝑏cos(𝑚𝜃)sin(𝑛𝜑), with θ∈[0,2π]𝜃∈[0,2𝜋] and φ∈[0,π],𝜑∈[0,𝜋], where a𝑎 and b𝑏 are positive numbers and m𝑚 and n𝑛 are positive integers, may be used in applied mathematics to model tumor growth. 1. Show that the “bumpy sphere” is contained inside a sphere of equation ρ=a+b.𝜌=𝑎+𝑏. Find the values of θ𝜃 and φ𝜑 at which the two surfaces intersect. 2. Use a CAS to graph the surface for a=14,𝑎=14, b=2,𝑏=2, m=4,𝑚=4, and n=6𝑛=6 along with sphere ρ=a+b.𝜌=𝑎+𝑏. 3. Find the equation of the intersection curve of the surface at b. with the cone φ=π12.𝜑=𝜋12. Graph the intersection curve in the plane of intersection. [Previous](https://openstax.org/books/calculus-volume-3/pages/2-6-quadric-surfaces)[Next](https://openstax.org/books/calculus-volume-3/pages/2-key-terms)  Win prizes by participating in research and discover more insights about yourself! [Explore Kinetic](https://openstax.org/kinetic/) [Order a print copy](https://amzn.to/3onqgUt) As an Amazon Associate we earn from qualifying purchases. Citation/Attribution Our mission is to improve educational access and learning for everyone. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. [Give today](https://openstax.org/give) and help us reach more students. 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Advanced Placement® and AP® are trademarks registered and/or owned by the College Board, which is not affiliated with, and does not endorse, this site. - [](https://www.facebook.com/openstax) - [](https://twitter.com/openstax) - [](https://www.linkedin.com/company/openstax) - [](https://www.instagram.com/openstax/) - [](http://www.rice.edu/) - Solution The radius of Earth is 4000 mi, so ρ=4000 The intersection of the prime meridian and the equator lies on the positive _x_-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83°, Because Columbus lies 40° north of the equator, it lies 50° south of the North Pole, so φ=50°. In spherical coordinates, Columbus lies at point (4000,−83°,50°). ### Converting among Spherical, Cylindrical, and Rectangular Coordinates Rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) and spherical coordinates (ρ,θ,φ)(𝜌,𝜃,𝜑) of a point are related as follows: xyzρ2tanθφ===and===ρsinφcosθρsinφsinθρcosφx2+y2+z2yxarccos(zx2+y2+z2√).These equations are used to convert fromspherical coordinates to rectangularcoordinates.These equations are used to convert fromrectangular coordinates to sphericalcoordinates.𝑥=𝜌sin𝜑cos𝜃These equations are used to convert from𝑦=𝜌sin𝜑sin𝜃spherical coordinates to rectangular𝑧=𝜌cos𝜑coordinates.and𝜌2=𝑥2+𝑦2+𝑧2These equations are used to convert fromtan𝜃=𝑦𝑥rectangular coordinates to spherical𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2).coordinates. If a point has cylindrical coordinates (r,θ,z),(𝑟,𝜃,𝑧), then these equations define the relationship between cylindrical and spherical coordinates. rθzρθφ===and===ρsinφθρcosφr2+z2−−−−−−√θarccos(zr2+z2√)These equations are used to convert fromspherical coordinates to cylindricalcoordinates.These equations are used to convert fromcylindrical coordinates to sphericalcoordinates. ![[Attachments/Pasted image 20240513010600.png]] ![[Attachments/Pasted image 20240513010606.png]] The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at [Figure 2.98](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_012), it is easy to see that r=ρsinφ.𝑟=𝜌sin𝜑. Then, looking at the triangle in the _xy_-plane with r𝑟 as its hypotenuse, we have x=rcosθ=ρsinφcosθ.𝑥=𝑟cos𝜃=𝜌sin𝜑cos𝜃. The derivation of the formula for y𝑦 is similar. [Figure 2.96](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_009) also shows ![[Attachments/Pasted image 20240513010702.png]] As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let c be a constant, and consider surfaces of the form ρ=c. Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate θ in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form θ=c are half-planes, as before. Last, consider surfaces of the form φ=c. The points on these surfaces are at a fixed angle from the z-axis and form a half-cone (Figure 2.99). ![[Attachments/Pasted image 20240513010812.png]] Converting from Spherical Coordinates Plot the point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) and express its location in both rectangular and cylindrical coordinates. $x=ρsinφcosθ=8sin(π6)cos(π3)=8(12)12=2$ $y=ρsinφsinθ=8sin(π6)sin(π3)=8(12)3√2=23–√$ $z=ρcosφ=8cos(π6)=8(3√2)=43–√$ ![[Attachments/Pasted image 20240513010925.png]] ![[Attachments/9e41783dadcb6e0a855d174584644195e1410f61.jpg]] Figure 2.100 The projection of the point in the _xy_-plane is 44 units from the origin. The line from the origin to the point’s projection forms an angle of π/3𝜋/3 with the positive _x_-axis. The point lies 43–√43 units above the _xy_-plane. The point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) has rectangular coordinates (2,23–√,43–√).(2,23,43). Finding the values in cylindrical coordinates is equally straightforward: ![[Attachments/Pasted image 20240513010957.png]] THEOREM 2.15 Conversion between Cylindrical and Cartesian Coordinates The rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) and the cylindrical coordinates (r,θ,z)(𝑟,𝜃,𝑧) of a point are related as follows: xyzr2tanθz===and===rcosθrsinθzx2+y2yxzThese equations are used to convert fromcylindrical coordinates to rectangularcoordinates.These equations are used to convert fromrectangular coordinates to cylindricalcoordinates. ![[Attachments/Pasted image 20240513011914.png]] As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation tanθ=yxtan𝜃=𝑦𝑥 has an infinite number of solutions. However, if we restrict θ𝜃 to values between 00 and 2π,2𝜋, then we can find a unique solution based on the quadrant of the _xy_-plane in which original point (x,y,z)(𝑥,𝑦,𝑧) is located. Note that if x=0,𝑥=0, then the value of θ𝜃 is either π2,3π2,𝜋2,3𝜋2, or 0,0, depending on the value of y.𝑦. Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point P𝑃 in the _xy_-plane with rectangular coordinates (x,y,0)(𝑥,𝑦,0) and with cylindrical coordinates (r,θ,0),(𝑟,𝜃,0), as shown in the following figure. ![[Attachments/Pasted image 20240513011942.png]] Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If c is a constant, then in rectangular coordinates, surfaces of the form x=c, y=c, or z=c are all planes. Planes of these forms are parallel to the yz-plane, the xz-plane, and the xy-plane, respectively. When we convert to cylindrical coordinates, the z-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form z=c are planes parallel to the xy-plane. Now, let’s think about surfaces of the form r=c. The points on these surfaces are at a fixed distance from the z-axis. In other words, these surfaces are vertical circular cylinders. Last, what about θ=c? The points on a surface of the form θ=c are at a fixed angle from the x-axis, which gives us a half-plane that starts at the z-axis ![[Attachments/Pasted image 20240513012003.png]] Figure 2.91 In rectangular coordinates, (a) surfaces of the form x=c𝑥=𝑐 are planes parallel to the _yz_-plane, (b) surfaces of the form y=c𝑦=𝑐 are planes parallel to the _xz_-plane, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the _xy_-plane. ![[Attachments/Pasted image 20240513012014.png]] Figure 2.92 In cylindrical coordinates, (a) surfaces of the form r=c𝑟=𝑐 are vertical cylinders of radius c,𝑐, (b) surfaces of the form θ=c𝜃=𝑐 are half-planes at angle c𝑐 from the _x_-axis, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the _xy_-plane. ### Converting from Rectangular to Cylindrical Coordinates Convert the rectangular coordinates (1,−3,5)(1,−3,5) to cylindrical coordinates. Solution Use the second set of equations from [Conversion between Cylindrical and Cartesian Coordinates](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#fs-id1163723500624) to translate from rectangular to cylindrical coordinates: $r2==x2+y2±12+(−3)2 √=±10−−√.𝑟2=𝑥2+𝑦2𝑟=±12+(−3)2=±10.$ We choose the positive square root, so r=10−−√.𝑟=10. Now, we apply the formula to find θ.𝜃. In this case, y𝑦 is negative and x𝑥 is positive, which means we must select the value of θ𝜃 between 3π23𝜋2 and 2π:2𝜋: $tanθ==y x=−31arctan(−3)+2π≈5.03rad.tan𝜃=𝑦𝑥=−31𝜃=arctan(−3)+2𝜋≈5.03rad.$ In this case, the _z_-coordinates are the same in both rectangular and cylindrical coordinates: z=5.𝑧=5. The point with rectangular coordinates (1,−3,5)(1,−3,5) has cylindrical coordinates approximately equal to (10−−√,5.03,5). ![[Attachments/Pasted image 20240513012233.png]] ### Convert point (−8,8,−7)(−8,8,−7) from Cartesian coordinates to cylindrical coordinates. The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation x2+y2=25𝑥2+𝑦2=25 in the Cartesian system can be represented by cylindrical equation r=5. ![[Attachments/Pasted image 20240513012303.png]] ## EXAMPLE 2.65 ### Identifying Surfaces in the Spherical Coordinate System Describe the surfaces with the given spherical equations. 1. θ=π3𝜃=𝜋3 2. φ=5π6𝜑=5𝜋6 3. ρ=6𝜌=6 4. ρ=sinθsinφ𝜌=sin𝜃sin𝜑 ### Solution 1. The variable θ𝜃 represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates (ρ,π3,φ)(𝜌,𝜋3,𝜑) lie on the plane that forms angle θ=π3𝜃=𝜋3 with the positive _x_-axis. Because ρ>0,𝜌>0, the surface described by equation θ=π3𝜃=𝜋3 is the half-plane shown in [Figure 2.101](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_016).  Figure 2.101 The surface described by equation θ=π3𝜃=𝜋3 is a half-plane. 2. Equation φ=5π6𝜑=5𝜋6 describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring 5π65𝜋6 rad with the positive _z_-axis. These points form a half-cone ([Figure 2.102](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_017)). Because there is only one value for φ𝜑 that is measured from the positive _z_-axis, we do not get the full cone (with two pieces).  Figure 2.102 The equation φ=5π6𝜑=5𝜋6 describes a cone. To find the equation in rectangular coordinates, use equation φ=arccos(zx2+y2+z2√).𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2). 5π6cos5π6−3√2343x24+3y24+3z243x24+3y24−z24======arccos(zx2+y2+z2√)zx2+y2+z2√zx2+y2+z2√z2x2+y2+z2z20.5𝜋6=arccos(𝑧𝑥2+𝑦2+𝑧2)cos5𝜋6=𝑧𝑥2+𝑦2+𝑧2−32=𝑧𝑥2+𝑦2+𝑧234=𝑧2𝑥2+𝑦2+𝑧23𝑥24+3𝑦24+3𝑧24=𝑧23𝑥24+3𝑦24−𝑧24=0. This is the equation of a cone centered on the _z_-axis. 3. Equation ρ=6𝜌=6 describes the set of all points 66 units away from the origin—a sphere with radius 66 ([Figure 2.103](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_018)).  Figure 2.103 Equation ρ=6𝜌=6 describes a sphere with radius 6.6. 4. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations y=ρsinφsinθ𝑦=𝜌sin𝜑sin𝜃 and ρ2=x2+y2+z2:𝜌2=𝑥2+𝑦2+𝑧2: ρ2x2+y2+z2x2+y2−y+z2x2+y2−y+14+z2x2+(y−12)2+z2======sinθsinφρsinθsinφy01414.Multiply both sides of the equation byρ.Substitute rectangular variables using the equations above.Subtractyfrom both sides of the equation. Complete the square.Rewrite the middle terms as a perfect square. $𝜌=sin𝜃sin𝜑𝜌2=𝜌sin𝜃sin𝜑$ Multiply both sides of the equation b $y𝜌.𝑥2+𝑦2+𝑧2=𝑦$ Substitute rectangular variables using the equations above. 𝑥2+𝑦2−𝑦+𝑧2=0 Subtract 𝑦f rom both sides of the equation. $𝑥2+𝑦2−𝑦+14+𝑧2=14$ Complete the square. $𝑥2+(𝑦−12)2+𝑧2=14.$ Rewrite the middle terms as a perfect square. The equation describes a sphere centered at point (0,12,0)(0,12,0) with radius 12.12. ## CHECKPOINT 2.59 Describe the surfaces defined by the following equations. 1. ρ=13𝜌=13 2. θ=2π3𝜃=2𝜋3 3. φ=π4𝜑=𝜋4 Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x2+y2+z2=c2𝑥2+𝑦2+𝑧2=𝑐2 has the simple equation ρ=c𝜌=𝑐 in spherical coordinates. In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in [Figure 2.104](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_019). Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 40004000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.  Figure 2.104 In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian. Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive _z_-axis. The prime meridian represents the trace of the surface as it intersects the _xz_-plane. The equator is the trace of the sphere intersecting the _xy_-plane. ## Geographical ![[Attachments/Pasted image 20240512204358.png]] ![[Attachments/Pasted image 20240512204557.png]] ![[Attachments/Pasted image 20240512204634.png]] ![[Attachments/Pasted image 20240512204648.png]] ![[Attachments/Pasted image 20240512204702.png]] ![[Attachments/Pasted image 20240512204719.png]] ![[Attachments/Pasted image 20240512204731.png]] ![[Attachments/Pasted image 20240512204743.png]] ![[Attachments/Pasted image 20240512204755.png]] ![[Attachments/Pasted image 20240512204806.png]] ![[Attachments/Pasted image 20240512204820.png]] ![[Attachments/Pasted image 20240512204831.png]] ![[Attachments/Pasted image 20240512204855.png]] ![[Attachments/Pasted image 20240512204907.png]] ![[Attachments/Pasted image 20240512204915.png]] ![[Attachments/Pasted image 20240512204928.png]] ![[Attachments/Pasted image 20240512204941.png]] ![[Attachments/Pasted image 20240512204954.png]] ### UTM Coordinate System ![[Attachments/Pasted image 20240512205010.png]] ![[Attachments/Pasted image 20240512223825.png]] ![[Attachments/Pasted image 20240512223831.png]] ![[Attachments/Pasted image 20240512223838.png]] ![[Attachments/Pasted image 20240512223857.png]] ![[Attachments/Pasted image 20240512223847.png]] ![[Attachments/Pasted image 20240512223907.png]] ![[Attachments/Pasted image 20240512223921.png]] ![[Attachments/Pasted image 20240512223930.png]] **Coordinate Transformations** A great deal of the practical side of celestial mechanics involves transforming observational quantities from one coordinate system to another. Thus it is appropriate that we discuss the manner in which this is done in general to find the rules that apply to the problems we will encounter in celestial mechanics. While within the framework of mathematics it is possible to define myriads of coordinate transformations, we shall concern ourselves with a special subset called linear transformations. Such coordinate transformations relate the coordinates in one frame to those in a second frame by means of a system of linear algebraic equations. Thus if a vector X r in one coordinate system has components Xj, in a primed-coordinate system a vector X' r to the same point will have components Xj given by i j ij j ![[Attachments/Pasted image 20240512210835.png]] ![[Attachments/Pasted image 20240512210841.png]] ![[Attachments/Pasted image 20240512210932.png]] ![[Attachments/Pasted image 20240514225850.png]] ### Circular Cylindrical Coordinates ![[Attachments/Coordinate systems.png]] ![[Attachments/Lesson-6-Polar-Cylindrical-and-Spherical-coordinates-15-2048.jpg]] ![[Attachments/Lesson-6-Polar-Cylindrical-and-Spherical-coordinates-12-2048.jpg]] ![[Attachments/coordinateandunitvector-150711070134-lva1-app6891-thumbnail.jpg]] ![[Attachments/cylinderandsphere-170320023614-thumbnail.jpg]] ![[Attachments/Pasted image 20240513031318.png]] ### More transforms practice: https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates ## Learning Objectives - 2.7.1 Convert from cylindrical to rectangular coordinates. - 2.7.2 Convert from rectangular to cylindrical coordinates. - 2.7.3 Convert from spherical to rectangular coordinates. - 2.7.4 Convert from rectangular to spherical coordinates. The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures. ## Cylindrical Coordinates When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. ## DEFINITION In the cylindrical coordinate system, a point in space ([Figure 2.89](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_001)) is represented by the ordered triple (r,θ,z),(𝑟,𝜃,𝑧), where - (r,θ)(𝑟,𝜃) are the polar coordinates of the point’s projection in the _xy_-plane - z𝑧 is the usual z-coordinate𝑧-coordinate in the Cartesian coordinate system  Figure 2.89 The right triangle lies in the _xy_-plane. The length of the hypotenuse is r𝑟 and θ𝜃 is the measure of the angle formed by the positive _x_-axis and the hypotenuse. The _z_-coordinate describes the location of the point above or below the _xy_-plane. In the _xy_-plane, the right triangle shown in [Figure 2.89](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_001) provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates. ## THEOREM 2.15 ### Conversion between Cylindrical and Cartesian Coordinates The rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) and the cylindrical coordinates (r,θ,z)(𝑟,𝜃,𝑧) of a point are related as follows: xyzr2tanθz===and===rcosθrsinθzx2+y2yxzThese equations are used to convert fromcylindrical coordinates to rectangularcoordinates.These equations are used to convert fromrectangular coordinates to cylindricalcoordinates.𝑥=𝑟cos𝜃These equations are used to convert from𝑦=𝑟sin𝜃cylindrical coordinates to rectangular𝑧=𝑧coordinates.and𝑟2=𝑥2+𝑦2These equations are used to convert fromtan𝜃=𝑦𝑥rectangular coordinates to cylindrical𝑧=𝑧coordinates. As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation tanθ=yxtan𝜃=𝑦𝑥 has an infinite number of solutions. However, if we restrict θ𝜃 to values between 00 and 2π,2𝜋, then we can find a unique solution based on the quadrant of the _xy_-plane in which original point (x,y,z)(𝑥,𝑦,𝑧) is located. Note that if x=0,𝑥=0, then the value of θ𝜃 is either π2,3π2,𝜋2,3𝜋2, or 0,0, depending on the value of y.𝑦. Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point P𝑃 in the _xy_-plane with rectangular coordinates (x,y,0)(𝑥,𝑦,0) and with cylindrical coordinates (r,θ,0),(𝑟,𝜃,0), as shown in the following figure.  Figure 2.90 The Pythagorean theorem provides equation r2=x2+y2.𝑟2=𝑥2+𝑦2. Right-triangle relationships tell us that x=rcosθ,𝑥=𝑟cos𝜃, y=rsinθ,𝑦=𝑟sin𝜃, and tanθ=y/x.tan𝜃=𝑦/𝑥. Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If c𝑐 is a constant, then in rectangular coordinates, surfaces of the form x=c,𝑥=𝑐, y=c,𝑦=𝑐, or z=c𝑧=𝑐 are all planes. Planes of these forms are parallel to the _yz_-plane, the _xz_-plane, and the _xy_-plane, respectively. When we convert to cylindrical coordinates, the _z_-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form z=c𝑧=𝑐 are planes parallel to the _xy_-plane. Now, let’s think about surfaces of the form r=c.𝑟=𝑐. The points on these surfaces are at a fixed distance from the _z_-axis. In other words, these surfaces are vertical circular cylinders. Last, what about θ=c?𝜃=𝑐? The points on a surface of the form θ=c𝜃=𝑐 are at a fixed angle from the _x_-axis, which gives us a half-plane that starts at the _z_-axis ([Figure 2.91](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_003) and [Figure 2.92](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_004)).  Figure 2.91 In rectangular coordinates, (a) surfaces of the form x=c𝑥=𝑐 are planes parallel to the _yz_-plane, (b) surfaces of the form y=c𝑦=𝑐 are planes parallel to the _xz_-plane, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the _xy_-plane.  Figure 2.92 In cylindrical coordinates, (a) surfaces of the form r=c𝑟=𝑐 are vertical cylinders of radius c,𝑐, (b) surfaces of the form θ=c𝜃=𝑐 are half-planes at angle c𝑐 from the _x_-axis, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the _xy_-plane. ## EXAMPLE 2.60 ### Converting from Cylindrical to Rectangular Coordinates Plot the point with cylindrical coordinates (4,2π3,−2)(4,2𝜋3,−2) and express its location in rectangular coordinates. ### Solution Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in [Conversion between Cylindrical and Cartesian Coordinates](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#fs-id1163723500624): xyz===rcosθ=4cos2π3=−2rsinθ=4sin2π3=23–√−2.𝑥=𝑟cos𝜃=4cos2𝜋3=−2𝑦=𝑟sin𝜃=4sin2𝜋3=23𝑧=−2. The point with cylindrical coordinates (4,2π3,−2)(4,2𝜋3,−2) has rectangular coordinates (−2,23–√,−2)(−2,23,−2) (see the following figure).  Figure 2.93 The projection of the point in the _xy_-plane is 4 units from the origin. The line from the origin to the point’s projection forms an angle of 2π32𝜋3 with the positive _x_-axis. The point lies 22 units below the _xy_-plane. ## CHECKPOINT 2.55 Point R𝑅 has cylindrical coordinates (5,π6,4)(5,𝜋6,4). Plot R𝑅 and describe its location in space using rectangular, or Cartesian, coordinates. If this process seems familiar, it is with good reason. This is exactly the same process that we followed in [Introduction to Parametric Equations and Polar Coordinates](https://openstax.org/books/calculus-volume-3/pages/1-introduction) to convert from polar coordinates to two-dimensional rectangular coordinates. ## EXAMPLE 2.61 ### Converting from Rectangular to Cylindrical Coordinates Convert the rectangular coordinates (1,−3,5)(1,−3,5) to cylindrical coordinates. ### Solution Use the second set of equations from [Conversion between Cylindrical and Cartesian Coordinates](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#fs-id1163723500624) to translate from rectangular to cylindrical coordinates: r2r==x2+y2±12+(−3)2−−−−−−−−−√=±10−−√.𝑟2=𝑥2+𝑦2𝑟=±12+(−3)2=±10. We choose the positive square root, so r=10−−√.𝑟=10. Now, we apply the formula to find θ.𝜃. In this case, y𝑦 is negative and x𝑥 is positive, which means we must select the value of θ𝜃 between 3π23𝜋2 and 2π:2𝜋: tanθθ==yx=−31arctan(−3)+2π≈5.03rad.tan𝜃=𝑦𝑥=−31𝜃=arctan(−3)+2𝜋≈5.03rad. In this case, the _z_-coordinates are the same in both rectangular and cylindrical coordinates: z=5.𝑧=5. The point with rectangular coordinates (1,−3,5)(1,−3,5) has cylindrical coordinates approximately equal to (10−−√,5.03,5).(10,5.03,5). ## CHECKPOINT 2.56 Convert point (−8,8,−7)(−8,8,−7) from Cartesian coordinates to cylindrical coordinates. The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation x2+y2=25𝑥2+𝑦2=25 in the Cartesian system can be represented by cylindrical equation r=5.𝑟=5. ## EXAMPLE 2.62 ### Identifying Surfaces in the Cylindrical Coordinate System Describe the surfaces with the given cylindrical equations. 1. θ=π4𝜃=𝜋4 2. r2+z2=9𝑟2+𝑧2=9 3. z=r𝑧=𝑟 ### Solution 1. When the angle θ𝜃 is held constant while r𝑟 and z𝑧 are allowed to vary, the result is a half-plane (see the following figure).  Figure 2.94 In polar coordinates, the equation θ=π/4𝜃=𝜋/4 describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane. 2. Substitute r2=x2+y2𝑟2=𝑥2+𝑦2 into equation r2+z2=9𝑟2+𝑧2=9 to express the rectangular form of the equation: x2+y2+z2=9.𝑥2+𝑦2+𝑧2=9. This equation describes a sphere centered at the origin with radius 33 (see the following figure).  Figure 2.95 The sphere centered at the origin with radius 33 can be described by the cylindrical equation r2+z2=9.𝑟2+𝑧2=9. 3. To describe the surface defined by equation z=r,𝑧=𝑟, is it useful to examine traces parallel to the _xy_-plane. For example, the trace in plane z=1𝑧=1 is circle r=1,𝑟=1, the trace in plane z=3𝑧=3 is circle r=3,𝑟=3, and so on. Each trace is a circle. As the value of z𝑧 increases, the radius of the circle also increases. The resulting surface is a cone (see the following figure).  Figure 2.96 The traces in planes parallel to the _xy_-plane are circles. The radius of the circles increases as z𝑧 increases. ## CHECKPOINT 2.57 Describe the surface with cylindrical equation r=6.𝑟=6. ## Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances (randz)(𝑟and𝑧) and an angle measure (θ).(𝜃). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates. ## DEFINITION In the spherical coordinate system, a point P𝑃 in space ([Figure 2.97](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_011)) is represented by the ordered triple (ρ,θ,φ)(𝜌,𝜃,𝜑) where - ρ𝜌 (the Greek letter rho) is the distance between P𝑃 and the origin (ρ≠0);(𝜌≠0); - θ𝜃 is the same angle used to describe the location in cylindrical coordinates; - φ𝜑 (the Greek letter phi) is the angle formed by the positive _z_-axis and line segment OP––––,𝑂𝑃—, where O𝑂 is the origin and 0≤φ≤π.0≤𝜑≤𝜋.  Figure 2.97 The relationship among spherical, rectangular, and cylindrical coordinates. By convention, the origin is represented as (0,0,0)(0,0,0) in spherical coordinates. ## THEOREM 2.16 ### Converting among Spherical, Cylindrical, and Rectangular Coordinates Rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) and spherical coordinates (ρ,θ,φ)(𝜌,𝜃,𝜑) of a point are related as follows: xyzρ2tanθφ===and===ρsinφcosθρsinφsinθρcosφx2+y2+z2yxarccos(zx2+y2+z2√).These equations are used to convert fromspherical coordinates to rectangularcoordinates.These equations are used to convert fromrectangular coordinates to sphericalcoordinates.𝑥=𝜌sin𝜑cos𝜃These equations are used to convert from𝑦=𝜌sin𝜑sin𝜃spherical coordinates to rectangular𝑧=𝜌cos𝜑coordinates.and𝜌2=𝑥2+𝑦2+𝑧2These equations are used to convert fromtan𝜃=𝑦𝑥rectangular coordinates to spherical𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2).coordinates. If a point has cylindrical coordinates (r,θ,z),(𝑟,𝜃,𝑧), then these equations define the relationship between cylindrical and spherical coordinates. rθzρθφ===and===ρsinφθρcosφr2+z2−−−−−−√θarccos(zr2+z2√)These equations are used to convert fromspherical coordinates to cylindricalcoordinates.These equations are used to convert fromcylindrical coordinates to sphericalcoordinates.𝑟=𝜌sin𝜑These equations are used to convert from𝜃=𝜃spherical coordinates to cylindrical𝑧=𝜌cos𝜑coordinates.and𝜌=𝑟2+𝑧2These equations are used to convert from𝜃=𝜃cylindrical coordinates to spherical𝜑=arccos(𝑧𝑟2+𝑧2)coordinates. The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at [Figure 2.98](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_012), it is easy to see that r=ρsinφ.𝑟=𝜌sin𝜑. Then, looking at the triangle in the _xy_-plane with r𝑟 as its hypotenuse, we have x=rcosθ=ρsinφcosθ.𝑥=𝑟cos𝜃=𝜌sin𝜑cos𝜃. The derivation of the formula for y𝑦 is similar. [Figure 2.96](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_009) also shows that ρ2=r2+z2=x2+y2+z2𝜌2=𝑟2+𝑧2=𝑥2+𝑦2+𝑧2 and z=ρcosφ.𝑧=𝜌cos𝜑. Solving this last equation for φ𝜑 and then substituting ρ=r2+z2−−−−−−√𝜌=𝑟2+𝑧2 (from the first equation) yields φ=arccos(zr2+z2√).𝜑=arccos(𝑧𝑟2+𝑧2). Also, note that, as before, we must be careful when using the formula tanθ=yxtan𝜃=𝑦𝑥 to choose the correct value of θ.𝜃.  Figure 2.98 The equations that convert from one system to another are derived from right-triangle relationships. As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let c𝑐 be a constant, and consider surfaces of the form ρ=c.𝜌=𝑐. Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate θ𝜃 in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form θ=c𝜃=𝑐 are half-planes, as before. Last, consider surfaces of the form φ=c.𝜑=𝑐. The points on these surfaces are at a fixed angle from the _z_-axis and form a half-cone ([Figure 2.99](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_013)).  Figure 2.99 In spherical coordinates, surfaces of the form ρ=c𝜌=𝑐 are spheres of radius ρ𝜌 (a), surfaces of the form θ=c𝜃=𝑐 are half-planes at an angle θ𝜃 from the _x_-axis (b), and surfaces of the form ϕ=c𝜙=𝑐 are half-cones at an angle ϕ𝜙 from the _z_-axis (c). ## EXAMPLE 2.63 ### Converting from Spherical Coordinates Plot the point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) and express its location in both rectangular and cylindrical coordinates. ### Solution Use the equations in [Converting among Spherical, Cylindrical, and Rectangular Coordinates](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#fs-id1163723844895) to translate between spherical and cylindrical coordinates ([Figure 2.100](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_014)): x=ρsinφcosθ=8sin(π6)cos(π3)=8(12)12=2y=ρsinφsinθ=8sin(π6)sin(π3)=8(12)3√2=23–√z=ρcosφ=8cos(π6)=8(3√2)=43–√.𝑥=𝜌sin𝜑cos𝜃=8sin(𝜋6)cos(𝜋3)=8(12)12=2𝑦=𝜌sin𝜑sin𝜃=8sin(𝜋6)sin(𝜋3)=8(12)32=23𝑧=𝜌cos𝜑=8cos(𝜋6)=8(32)=43.  Figure 2.100 The projection of the point in the _xy_-plane is 44 units from the origin. The line from the origin to the point’s projection forms an angle of π/3𝜋/3 with the positive _x_-axis. The point lies 43–√43 units above the _xy_-plane. The point with spherical coordinates (8,π3,π6)(8,𝜋3,𝜋6) has rectangular coordinates (2,23–√,43–√).(2,23,43). Finding the values in cylindrical coordinates is equally straightforward: rθz===ρsinφ=8sinπ6=4θρcosφ=8cosπ6=43–√.𝑟=𝜌sin𝜑=8sin𝜋6=4𝜃=𝜃𝑧=𝜌cos𝜑=8cos𝜋6=43. Thus, cylindrical coordinates for the point are (4,π3,43–√).(4,𝜋3,43). ## CHECKPOINT 2.58 Plot the point with spherical coordinates (2,−5π6,π6)(2,−5𝜋6,𝜋6) and describe its location in both rectangular and cylindrical coordinates. ## EXAMPLE 2.64 ### Converting from Rectangular Coordinates Convert the rectangular coordinates (−1,1,6–√)(−1,1,6) to both spherical and cylindrical coordinates. ### Solution Start by converting from rectangular to spherical coordinates: ρ2ρ==x2+y2+z2=(−1)2+12+(6–√)2=822–√tanθθ==1−1arctan(−1)=3π4.𝜌2=𝑥2+𝑦2+𝑧2=(−1)2+12+(6)2=8𝜌=22tan𝜃=1−1𝜃=arctan(−1)=3𝜋4. Because (x,y)=(−1,1),(𝑥,𝑦)=(−1,1), then the correct choice for θ𝜃 is 3π4.3𝜋4. There are actually two ways to identify φ.𝜑. We can use the equation φ=arccos(zx2+y2+z2√).𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2). A more simple approach, however, is to use equation z=ρcosφ.𝑧=𝜌cos𝜑. We know that z=6–√𝑧=6 and ρ=22–√,𝜌=22, so 6–√=22–√cosφ,socosφ=6–√22–√=3–√26=22cos𝜑,socos𝜑=622=32 and therefore φ=π6.𝜑=𝜋6. The spherical coordinates of the point are (22–√,3π4,π6).(22,3𝜋4,𝜋6). To find the cylindrical coordinates for the point, we need only find r:𝑟: r=ρsinφ=22–√sin(π6)=2–√.𝑟=𝜌sin𝜑=22sin(𝜋6)=2. The cylindrical coordinates for the point are (2–√,3π4,6–√).(2,3𝜋4,6). ## EXAMPLE 2.65 ### Identifying Surfaces in the Spherical Coordinate System Describe the surfaces with the given spherical equations. 1. θ=π3𝜃=𝜋3 2. φ=5π6𝜑=5𝜋6 3. ρ=6𝜌=6 4. ρ=sinθsinφ𝜌=sin𝜃sin𝜑 ### Solution 1. The variable θ𝜃 represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates (ρ,π3,φ)(𝜌,𝜋3,𝜑) lie on the plane that forms angle θ=π3𝜃=𝜋3 with the positive _x_-axis. Because ρ>0,𝜌>0, the surface described by equation θ=π3𝜃=𝜋3 is the half-plane shown in [Figure 2.101](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_016).  Figure 2.101 The surface described by equation θ=π3𝜃=𝜋3 is a half-plane. 2. Equation φ=5π6𝜑=5𝜋6 describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring 5π65𝜋6 rad with the positive _z_-axis. These points form a half-cone ([Figure 2.102](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_017)). Because there is only one value for φ𝜑 that is measured from the positive _z_-axis, we do not get the full cone (with two pieces).  Figure 2.102 The equation φ=5π6𝜑=5𝜋6 describes a cone. To find the equation in rectangular coordinates, use equation φ=arccos(zx2+y2+z2√).𝜑=arccos(𝑧𝑥2+𝑦2+𝑧2). 5π6cos5π6−3√2343x24+3y24+3z243x24+3y24−z24======arccos(zx2+y2+z2√)zx2+y2+z2√zx2+y2+z2√z2x2+y2+z2z20.5𝜋6=arccos(𝑧𝑥2+𝑦2+𝑧2)cos5𝜋6=𝑧𝑥2+𝑦2+𝑧2−32=𝑧𝑥2+𝑦2+𝑧234=𝑧2𝑥2+𝑦2+𝑧23𝑥24+3𝑦24+3𝑧24=𝑧23𝑥24+3𝑦24−𝑧24=0. This is the equation of a cone centered on the _z_-axis. 3. Equation ρ=6𝜌=6 describes the set of all points 66 units away from the origin—a sphere with radius 66 ([Figure 2.103](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_018)).  Figure 2.103 Equation ρ=6𝜌=6 describes a sphere with radius 6.6. 4. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations y=ρsinφsinθ𝑦=𝜌sin𝜑sin𝜃 and ρ2=x2+y2+z2:𝜌2=𝑥2+𝑦2+𝑧2: ρρ2x2+y2+z2x2+y2−y+z2x2+y2−y+14+z2x2+(y−12)2+z2======sinθsinφρsinθsinφy01414.Multiply both sides of the equation byρ.Substitute rectangular variables using the equations above.Subtractyfrom both sides of the equation.Complete the square.Rewrite the middle terms as a perfect square.𝜌=sin𝜃sin𝜑𝜌2=𝜌sin𝜃sin𝜑Multiply both sides of the equation by𝜌.𝑥2+𝑦2+𝑧2=𝑦Substitute rectangular variables using the equations above.𝑥2+𝑦2−𝑦+𝑧2=0Subtract𝑦from both sides of the equation.𝑥2+𝑦2−𝑦+14+𝑧2=14Complete the square.𝑥2+(𝑦−12)2+𝑧2=14.Rewrite the middle terms as a perfect square. The equation describes a sphere centered at point (0,12,0)(0,12,0) with radius 12.12. ## CHECKPOINT 2.59 Describe the surfaces defined by the following equations. 1. ρ=13𝜌=13 2. θ=2π3𝜃=2𝜋3 3. φ=π4𝜑=𝜋4 Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x2+y2+z2=c2𝑥2+𝑦2+𝑧2=𝑐2 has the simple equation ρ=c𝜌=𝑐 in spherical coordinates. In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in [Figure 2.104](https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates#CNX_Calc_Figure_12_07_019). Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 40004000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.  Figure 2.104 In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian. Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive _z_-axis. The prime meridian represents the trace of the surface as it intersects the _xz_-plane. The equator is the trace of the sphere intersecting the _xy_-plane. ## EXAMPLE 2.66 ### Converting Latitude and Longitude to Spherical Coordinates The latitude of Columbus, Ohio, is 40°40° N and the longitude is 83°83° W, which means that Columbus is 40°40° north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40°.40°. In the same way, measuring from the prime meridian, Columbus lies 83°83° to the west. Express the location of Columbus in spherical coordinates. ### Solution The radius of Earth is 40004000 mi, so ρ=4000.𝜌=4000. The intersection of the prime meridian and the equator lies on the positive _x_-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83°,𝜃=−83°, Because Columbus lies 40°40° north of the equator, it lies 50°50° south of the North Pole, so φ=50°.𝜑=50°. In spherical coordinates, Columbus lies at point (4000,−83°,50°).(4000,−83°,50°). ## CHECKPOINT 2.61 Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?  How should we orient the coordinate axes? [Skip to main content](https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates#elm-main-content "Press enter to skip to the main content") https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates ![[Attachments/Pasted image 20240513023213.png]] ![[Attachments/Pasted image 20240513023237.png]] ## Cylindrical Coordinates https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the  cylindrical coordinate system . In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. ##### Definition: The  Cylindrical Coordinate System In the  cylindrical coordinate system **,** a point in space (Figure 12.7.112.7.1) is represented by the ordered triple (r,θ,z)(𝑟,θ,𝑧), where - (r,θ)(𝑟,θ) are the polar coordinates of the point’s projection in the xy𝑥𝑦-plane - z𝑧 is the usual z𝑧-**coordinate** in the Cartesian coordinate system  Figure 12.7.112.7.1: The right triangle lies in the xy𝑥𝑦-plane. The length of the hypotenuse is r𝑟 and θθ is the measure of the angle formed by the positive x𝑥-axis and the hypotenuse. The z𝑧-coordinate describes the location of the point above or below the xy𝑥𝑦-plane. In the xy𝑥𝑦-plane, the right triangle shown in Figure 12.7.112.7.1 provides the key to  transformation  between cylindrical and Cartesian, or rectangular, coordinates. ##### Conversion between Cylindrical and Cartesian Coordinates The rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧) and the cylindrical coordinates (r,θ,z)(𝑟,θ,𝑧) of a point are related as follows: These equations are used to convert from cylindrical coordinates to rectangular coordinates. - x=rcosθ𝑥=𝑟cos⁡θ - y=rsinθ𝑦=𝑟sin⁡θ - z=z𝑧=𝑧 These equations are used to convert from rectangular coordinates to cylindrical coordinates 1. r2=x2+y2𝑟2=𝑥2+𝑦2 2. tanθ=yxtan⁡θ=𝑦𝑥 3. z=z𝑧=𝑧 As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation tanθ=yxtan⁡θ=𝑦𝑥 has an infinite number of solutions. However, if we restrict θθ to values between 00 and 2π2π, then we can find a unique solution based on the quadrant of the xy𝑥𝑦-plane in which original point (x,y,z)(𝑥,𝑦,𝑧) is located. Note that if x=0𝑥=0, then the value of θθ is either π2,3π2,π2,3π2, or 00, depending on the value of y𝑦. Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point P𝑃 in the xy𝑥𝑦-plane with rectangular coordinates (x,y,0)(𝑥,𝑦,0) and with cylindrical coordinates (r,θ,0)(𝑟,θ,0), as shown in Figure 12.7.212.7.2.  Figure 12.7.212.7.2: The Pythagorean theorem provides equation r2=x2+y2𝑟2=𝑥2+𝑦2. Right-triangle relationships tell us that x=rcosθ,y=rsinθ,𝑥=𝑟cos⁡θ,𝑦=𝑟sin⁡θ, and tanθ=y/x.tan⁡θ=𝑦/𝑥. Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If c𝑐 is a constant, then in rectangular coordinates, surfaces of the form x=c,y=c,𝑥=𝑐,𝑦=𝑐, or z=c𝑧=𝑐 are all planes. Planes of these forms are parallel to the yz𝑦𝑧-plane, the xz𝑥𝑧-plane, and the xy𝑥𝑦-plane, respectively. When we convert to cylindrical coordinates, the z𝑧-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form z=c𝑧=𝑐 are planes parallel to the xy𝑥𝑦-plane. Now, let’s think about surfaces of the form r=c𝑟=𝑐. The points on these surfaces are at a fixed distance from the z𝑧-axis. In other words, these surfaces are vertical circular cylinders. Last, what about θ=cθ=𝑐? The points on a  surface  of the form θ=cθ=𝑐 are at a fixed angle from the x𝑥-axis, which gives us a half-plane that starts at the z𝑧-axis (Figures 12.7.312.7.3 and 12.7.412.7.4).  Figure 12.7.312.7.3: In rectangular coordinates, (a) surfaces of the form x=c𝑥=𝑐 are planes parallel to the yz𝑦𝑧-plane, (b) surfaces of the form y=c𝑦=𝑐 are planes parallel to the xz𝑥𝑧-plane, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the xy𝑥𝑦-plane.  Figure 12.7.412.7.4: In cylindrical coordinates, (a) surfaces of the form r=c𝑟=𝑐 are vertical cylinders of radius r𝑟, (b) surfaces of the form θ=cθ=𝑐 are half-planes at angle θθ from the x𝑥-axis, and (c) surfaces of the form z=c𝑧=𝑐 are planes parallel to the xy𝑥𝑦-plane. ##### Example 12.7.112.7.1: Converting from Cylindrical to Rectangular Coordinates Plot the point with cylindrical coordinates (4,2π3,−2)(4,2π3,−2) and express its location in rectangular coordinates. **Solution** Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in Conversion between Cylindrical and Cartesian Coordinates: xyz=rcosθ=4cos2π3=−2=rsinθ=4sin2π3=23–√=−2.𝑥=𝑟cos⁡θ=4cos⁡2π3=−2𝑦=𝑟sin⁡θ=4sin⁡2π3=23𝑧=−2. The point with cylindrical coordinates (4,2π3,−2)(4,2π3,−2) has rectangular coordinates (−2,23–√,−2)(−2,23,−2) (Figure 12.7.512.7.5).  Figure 12.7.512.7.5: The projection of the point in the xy𝑥𝑦-plane is 4 units from the origin. The line from the origin to the point’s projection forms an angle of 2π32π3 with the positive x𝑥-axis. The point lies 22 units below the xy𝑥𝑦-plane. ##### Exercise 12.7.112.7.1 Point R𝑅 has cylindrical coordinates (5,π6,4)(5,π6,4). Plot R𝑅 and describe its location in space using rectangular, or Cartesian, coordinates. **Hint** **Answer** If this process seems familiar, it is with good reason. This is exactly the same process that we followed in Introduction to  Parametric Equations  and Polar Coordinates to convert from polar coordinates to two-dimensional rectangular coordinates. ##### Example 12.7.212.7.2: Converting from Rectangular to Cylindrical Coordinates Convert the rectangular coordinates (1,−3,5)(1,−3,5) to cylindrical coordinates. **Solution** Use the second set of equations from Conversion between Cylindrical and Cartesian Coordinates to translate from rectangular to cylindrical coordinates: r2r=x2+y2=±12+(−3)2−−−−−−−−−√=±10−−√.𝑟2=𝑥2+𝑦2𝑟=±12+(−3)2=±10. We choose the positive square root, so r=10−−√𝑟=10.Now, we apply the formula to find θθ. In this case, y𝑦 is negative and x𝑥 is positive, which means we must select the value of θθ between 3π23π2 and 2π2π: tanθθ=yx=arctan(−3)=−31≈5.03rad.tan⁡θ=𝑦𝑥=−31θ=arctan⁡(−3)≈5.03rad. In this case, the _z_-coordinates are the same in both rectangular and cylindrical coordinates: z=5.𝑧=5. The point with rectangular coordinates (1,−3,5)(1,−3,5) has cylindrical coordinates approximately equal to (10−−√,5.03,5).(10,5.03,5). ##### Exercise 12.7.212.7.2 Convert point (−8,8,−7)(−8,8,−7) from Cartesian coordinates to cylindrical coordinates. **Hint** **Answer** The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the  cylinder  described by equation x2+y2=25𝑥2+𝑦2=25 in the Cartesian system can be represented by cylindrical equation r=5𝑟=5. ##### Example 12.7.312.7.3: Identifying Surfaces in the  Cylindrical Coordinate System Describe the surfaces with the given cylindrical equations. 1. θ=π4θ=π4 2. r2+z2=9𝑟2+𝑧2=9 3. z=r𝑧=𝑟 **Solution** a. When the angle θθ is held constant while r𝑟 and z𝑧 are allowed to vary, the result is a half-plane (Figure 12.7.612.7.6).  Figure 12.7.612.7.6: In polar coordinates, the equation θ=π/4θ=π/4 describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane. b. Substitute r2=x2+y2𝑟2=𝑥2+𝑦2 into equation r2+z2=9𝑟2+𝑧2=9 to express the rectangular form of the equation: x2+y2+z2=9𝑥2+𝑦2+𝑧2=9. This equation describes a  sphere  centered at the origin with radius **3** (Figure 12.7.712.7.7).  Figure 12.7.712.7.7: The  sphere  centered at the origin with radius **3** can be described by the cylindrical equation r2+z2=9𝑟2+𝑧2=9. c. To describe the  surface  defined by equation z=r𝑧=𝑟, is it useful to examine traces parallel to the xy𝑥𝑦-plane. For example, the  trace  in plane z=1𝑧=1 is circle r=1𝑟=1, the  trace  in plane z=3𝑧=3 is circle r=3𝑟=3, and so on. Each  trace  is a circle. As the value of z𝑧 increases, the radius of the circle also increases. The resulting  surface  is a cone (Figure 12.7.812.7.8).  Figure 12.7.812.7.8: The traces in planes parallel to the xy𝑥𝑦-plane are circles. The radius of the circles increases as z𝑧 increases. ##### Exercise 12.7.312.7.3 Describe the  surface  with cylindrical equation r=6𝑟=6. **Hint** **Answer** The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the  cylinder  described by equation x2+y2=25𝑥2+𝑦2=25 in the Cartesian system can be represented by cylindrical equation r=5𝑟=5. ##### Example 12.7.312.7.3: Identifying Surfaces in the  Cylindrical Coordinate System Describe the surfaces with the given cylindrical equations. 1. θ=π4θ=π4 2. r2+z2=9𝑟2+𝑧2=9 3. z=r𝑧=𝑟 **Solution** a. When the angle θθ is held constant while r𝑟 and z𝑧 are allowed to vary, the result is a half-plane (Figure 12.7.612.7.6).  Figure 12.7.612.7.6: In polar coordinates, the equation θ=π/4θ=π/4 describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane. b. Substitute r2=x2+y2𝑟2=𝑥2+𝑦2 into equation r2+z2=9𝑟2+𝑧2=9 to express the rectangular form of the equation: x2+y2+z2=9𝑥2+𝑦2+𝑧2=9. This equation describes a  sphere  centered at the origin with radius **3** (Figure 12.7.712.7.7).  Figure 12.7.712.7.7: The  sphere  centered at the origin with radius **3** can be described by the cylindrical equation r2+z2=9𝑟2+𝑧2=9. c. To describe the  surface  defined by equation z=r𝑧=𝑟, is it useful to examine traces parallel to the xy𝑥𝑦-plane. For example, the  trace  in plane z=1𝑧=1 is circle r=1𝑟=1, the  trace  in plane z=3𝑧=3 is circle r=3𝑟=3, and so on. Each  trace  is a circle. As the value of z𝑧 increases, the radius of the circle also increases. The resulting  surface  is a cone (Figure 12.7.812.7.8).   ## Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the  cylindrical coordinate system , the location of a point in space is described using two distances (r(𝑟 and z)𝑧) and an angle measure (θ)(θ). In the  spherical coordinate system , we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a  sphere , just as cylindrical coordinates make it easy to describe a  cylinder . Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates. ##### Definition:  spherical coordinate system In the  _spherical coordinate system_ , a point P𝑃 in space (Figure 12.7.912.7.9) is represented by the ordered triple (ρ,θ,φ)(ρ,θ,φ) where - ρρ (the Greek letter rho) is the distance between P𝑃 and the origin (ρ≠0);(ρ≠0); - θθ is the same angle used to describe the location in cylindrical coordinates; - φφ (the Greek letter phi) is the angle formed by the positive z𝑧-axis and line segment OP¯¯¯¯¯¯¯¯𝑂𝑃¯, where O𝑂 is the origin and 0≤φ≤π.0≤φ≤π.  Figure 12.7.912.7.9: The relationship among spherical, rectangular, and cylindrical coordinates. By convention, the origin is represented as (0,0,0)(0,0,0) in spherical coordinates. ##### HOWTO: Converting among Spherical, Cylindrical, and Rectangular Coordinates Rectangular coordinates (x,y,z)(𝑥,𝑦,𝑧), cylindrical coordinates (r,θ,z),(𝑟,θ,𝑧), and spherical coordinates (ρ,θ,φ)(ρ,θ,φ) of a point are related as follows: **Convert from spherical coordinates to rectangular coordinates** These equations are used to convert from spherical coordinates to rectangular coordinates. - x=ρsinφcosθ𝑥=ρsin⁡φcos⁡θ - y=ρsinφsinθ𝑦=ρsin⁡φsin⁡θ - z=ρcosφ𝑧=ρcos⁡φ **Convert from rectangular coordinates to spherical coordinates** These equations are used to convert from rectangular coordinates to spherical coordinates. - ρ2=x2+y2+z2ρ2=𝑥2+𝑦2+𝑧2 - tanθ=yxtan⁡θ=𝑦𝑥 - φ=arccos(zx2+y2+z2−−−−−−−−−−√).φ=arccos⁡(𝑧𝑥2+𝑦2+𝑧2). **Convert from spherical coordinates to cylindrical coordinates** These equations are used to convert from spherical coordinates to cylindrical coordinates. - r=ρsinφ𝑟=ρsin⁡φ - θ=θθ=θ - z=ρcosφ𝑧=ρcos⁡φ **Convert from cylindrical coordinates to spherical coordinates** These equations are used to convert from cylindrical coordinates to spherical coordinates. - ρ=r2+z2−−−−−−√ρ=𝑟2+𝑧2 - θ=θθ=θ - φ=arccos(zr2+z2−−−−−−√)φ=arccos⁡(𝑧𝑟2+𝑧2) The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at Figure 12.7.1012.7.10, it is easy to see that r=ρsinφ𝑟=ρsin⁡φ. Then, looking at the triangle in the xy𝑥𝑦-plane with r as its hypotenuse, we have x=rcosθ=ρsinφcosθ𝑥=𝑟cos⁡θ=ρsin⁡φcos⁡θ. The derivation of the formula for y𝑦 is similar. Figure 12.7.1012.7.10 also shows that ρ2=r2+z2=x2+y2+z2ρ2=𝑟2+𝑧2=𝑥2+𝑦2+𝑧2 and z=ρcosφ𝑧=ρcos⁡φ. Solving this last equation for φφ and then substituting ρ=r2+z2−−−−−−√ρ=𝑟2+𝑧2 (from the first equation) yields φ=arccos(zr2+z2−−−−−−√)φ=arccos⁡(𝑧𝑟2+𝑧2). Also, note that, as before, we must be careful when using the formula tanθ=yxtan⁡θ=𝑦𝑥 to choose the correct value of θθ.  Figure 12.7.1012.7.10: The equations that convert from one system to another are derived from right-triangle relationships. As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let c𝑐 be a constant, and consider surfaces of the form ρ=cρ=𝑐. Points on these surfaces are at a fixed distance from the origin and form a  sphere . The coordinate θθ in the  spherical coordinate system  is the same as in the  cylindrical coordinate system , so surfaces of the form θ=cθ=𝑐 are half-planes, as before. Last, consider surfaces of the form φ=cφ=𝑐. The points on these surfaces are at a fixed angle from the z𝑧-axis and form a half-cone (Figure 12.7.1112.7.11).  Figure 12.7.1112.7.11: In spherical coordinates, surfaces of the form ρ=cρ=𝑐 are spheres of radius ρρ (a), surfaces of the form θ=cθ=𝑐 are half-planes at an angle θθ from the x𝑥-axis (b), and surfaces of the form ϕ=cϕ=𝑐 are half-cones at an angle ϕϕ from the z𝑧-axis (c). ##### Example 12.7.412.7.4: Converting from Spherical Coordinates Plot the point with spherical coordinates (8,π3,π6)(8,π3,π6) and express its location in both rectangular and cylindrical coordinates. **Solution** Use the equations in Converting among Spherical, Cylindrical, and Rectangular Coordinates to translate between spherical and cylindrical coordinates (Figure 12.7.1212.7.12): xyz=ρsinφcosθ=8sin(π6)cos(π3)=8(12)12=2=ρsinφsinθ=8sin(π6)sin(π3)=8(12)3–√2=23–√=ρcosφ=8cos(π6)=8(3–√2)=43–√𝑥=ρsin⁡φcos⁡θ=8sin⁡(π6)cos⁡(π3)=8(12)12=2𝑦=ρsin⁡φsin⁡θ=8sin⁡(π6)sin⁡(π3)=8(12)32=23𝑧=ρcos⁡φ=8cos⁡(π6)=8(32)=43  Figure 12.7.1212.7.12: The projection of the point in the xy𝑥𝑦-plane is 44 units from the origin. The line from the origin to the point’s projection forms an angle of π/3π/3 with the positive x𝑥-axis. The point lies 43–√43 units above the xy𝑥𝑦-plane. The point with spherical coordinates (8,π3,π6)(8,π3,π6) has rectangular coordinates (2,23–√,43–√).(2,23,43). Finding the values in cylindrical coordinates is equally straightforward: rθz=ρsinφ=8sinπ6=4=θ=ρcosφ=8cosπ6=43–√.𝑟=ρsin⁡φ=8sin⁡π6=4θ=θ𝑧=ρcos⁡φ=8cos⁡π6=43. Thus, cylindrical coordinates for the point are (4,π3,43–√)(4,π3,43). ##### Exercise 12.7.412.7.4 Plot the point with spherical coordinates (2,−5π6,π6)(2,−5π6,π6) and describe its location in both rectangular and cylindrical coordinates. **Hint** **Answer** ##### Example 12.7.512.7.5: Converting from Rectangular Coordinates Convert the rectangular coordinates (−1,1,6–√)(−1,1,6) to both spherical and cylindrical coordinates. **Solution** Start by converting from rectangular to spherical coordinates: ρ2tanθρ=x2+y2+z2=(−1)2+12+(6–√)2=8=1−1=22–√ and θ=arctan(−1)=3π4.ρ2=𝑥2+𝑦2+𝑧2=(−1)2+12+(6)2=8tan⁡θ=1−1ρ=22 and θ=arctan⁡(−1)=3π4. Because (x,y)=(−1,1)(𝑥,𝑦)=(−1,1), then the correct choice for θθ is 3π43π4. There are actually two ways to identify φφ. We can use the equation φ=arccos(zx2+y2+z2−−−−−−−−−−√)φ=arccos⁡(𝑧𝑥2+𝑦2+𝑧2). A more simple approach, however, is to use equation z=ρcosφ.𝑧=ρcos⁡φ. We know that z=6–√𝑧=6 and ρ=22–√ρ=22, so 6–√=22–√cosφ,6=22cos⁡φ, so cosφ=6–√22–√=3–√2cos⁡φ=622=32 and therefore φ=π6φ=π6. The spherical coordinates of the point are (22–√,3π4,π6).(22,3π4,π6). To find the cylindrical coordinates for the point, we need only find r𝑟: r=ρsinφ=22–√sin(π6)=2–√.𝑟=ρsin⁡φ=22sin⁡(π6)=2. The cylindrical coordinates for the point are (2–√,3π4,6–√)(2,3π4,6). ##### Example 12.7.612.7.6: Identifying Surfaces in the  Spherical Coordinate System Describe the surfaces with the given spherical equations. 1. θ=π3θ=π3 2. φ=5π6φ=5π6 3. ρ=6ρ=6 4. ρ=sinθsinφρ=sin⁡θsin⁡φ **Solution** a. The variable θθ represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates (ρ,π3,φ)(ρ,π3,φ) lie on the plane that forms angle θ=π3θ=π3 with the positive x𝑥-axis. Because ρ>0ρ>0, the  surface  described by equation θ=π3θ=π3 is the half-plane shown in Figure 12.7.1312.7.13.  Figure 12.7.1312.7.13: The  surface  described by equation θ=π3θ=π3 is a half-plane. b. Equation φ=5π6φ=5π6 describes all points in the  spherical coordinate system  that lie on a line from the origin forming an angle measuring 5π65π6 rad with the positive z𝑧-axis. These points form a half-cone (Figure 12.7.1412.7.14). Because there is only one value for φφ that is measured from the positive z𝑧-axis, we do not get the full cone (with two pieces).  Figure 12.7.1412.7.14: The equation φ=5π6φ=5π6 describes a cone. To find the equation in rectangular coordinates, use equation φ=arccos(zx2+y2+z2−−−−−−−−−−√).φ=arccos⁡(𝑧𝑥2+𝑦2+𝑧2). 5π6cos5π6−3–√2343x24+3y24+3z243x24+3y24−z24=arccos(zx2+y2+z2−−−−−−−−−−√)=zx2+y2+z2−−−−−−−−−−√=zx2+y2+z2−−−−−−−−−−√=z2x2+y2+z2=z2=0.5π6=arccos⁡(𝑧𝑥2+𝑦2+𝑧2)cos⁡5π6=𝑧𝑥2+𝑦2+𝑧2−32=𝑧𝑥2+𝑦2+𝑧234=𝑧2𝑥2+𝑦2+𝑧23𝑥24+3𝑦24+3𝑧24=𝑧23𝑥24+3𝑦24−𝑧24=0. This is the equation of a cone centered on the z𝑧-axis. c. Equation ρ=6ρ=6 describes the set of all points 66 units away from the origin—a  sphere  with radius 66 (Figure 12.7.1512.7.15).  Figure 12.7.1512.7.15: Equation ρ=6ρ=6 describes a  sphere  with radius 66. d. To identify this  surface , convert the equation from spherical to rectangular coordinates, using equations y=ρsinφsinθ𝑦=ρsin⁡φsin⁡θ and ρ2=x2+y2+z2:ρ2=𝑥2+𝑦2+𝑧2: ρ=sinθsinφρ=sin⁡θsin⁡φ ρ2=ρsinθsinφρ2=ρsin⁡θsin⁡φ Multiply both sides of the equation by ρρ. x2+y2+z2=y𝑥2+𝑦2+𝑧2=𝑦 Substitute rectangular variables using the equations above. x2+y2−y+z2=0𝑥2+𝑦2−𝑦+𝑧2=0 Subtract y𝑦 from both sides of the equation. x2+y2−y+14+z2=14𝑥2+𝑦2−𝑦+14+𝑧2=14 Complete the square. x2+(y−12)2+z2=14𝑥2+(𝑦−12)2+𝑧2=14. Rewrite the middle terms as a perfect square. The equation describes a  sphere  centered at point (0,12,0)(0,12,0) with radius 1212. ##### Exercise 12.7.512.7.5 Describe the surfaces defined by the following equations. 1. ρ=13ρ=13 2. θ=2π3θ=2π3 3. φ=π4φ=π4 **Hint** **Answer a** **Answer b** **Answer c** Spherical coordinates are useful in analyzing systems that have some  degree  of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A  sphere  that has Cartesian equation x2+y2+z2=c2𝑥2+𝑦2+𝑧2=𝑐2 has the simple equation ρ=cρ=𝑐 in spherical coordinates. In geography, latitude and longitude are used to describe locations on Earth’s  surface , as shown in Figure 12.7.1612.7.16. Although the shape of Earth is not a perfect  sphere , we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a  sphere  with radius 40004000 mi. We express angle measures in degrees rather than  radians  because latitude and longitude are measured in degrees.  Figure 12.7.1612.7.16: In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian. Let the center of Earth be the center of the  sphere , with the ray from the center through the North  Pole  representing the positive z𝑧-axis. The prime meridian represents the  trace  of the  surface  as it intersects the xz𝑥𝑧-plane. The equator is the  trace  of the  sphere  intersecting the xy𝑥𝑦-plane. ##### Example 12.7.712.7.7: Converting Latitude and Longitude to Spherical Coordinates The latitude of Columbus, Ohio, is 40°40° N and the longitude is 83°83° W, which means that Columbus is 40°40° north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40°40°. In the same way, measuring from the prime meridian, Columbus lies 83°83° to the west. Express the location of Columbus in spherical coordinates. **Solution** The radius of Earth is 40004000mi, so ρ=4000ρ=4000. The intersection of the prime meridian and the equator lies on the positive x𝑥-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83°θ=−83°, Because Columbus lies 40°40° north of the equator, it lies 50°50° south of the North  Pole , so φ=50°φ=50°. In spherical coordinates, Columbus lies at point (4000,−83°,50°).(4000,−83°,50°). ##### Exercise 12.7.612.7.6 Sydney, Australia is at 34°S34°𝑆 and 151°E.151°𝐸. Express Sydney’s location in spherical coordinates. **Hint** **Answer** Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one. ##### Example 12.7.812.7.8: Choosing the Best Coordinate System In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an  orientation  that makes sense in the context of the problem. Note: There is not enough information to set up or solve these problems; we simply select the coordinate system (Figure 12.7.1712.7.17). 1. Find the center of gravity of a bowling ball. 2. Determine the velocity of a submarine subjected to an ocean current. 3. Calculate the pressure in a conical water tank. 4. Find the volume of oil flowing through a pipeline. 5. Determine the amount of leather required to make a football.  Figure 12.7.1712.7.17: (credit: (a) modification of  work  by scl hua, Wikimedia, (b) modification of  work  by DVIDSHUB, Flickr, (c) modification of  work  by Michael Malak, Wikimedia, (d) modification of  work  by Sean Mack, Wikimedia, (e) modification of  work  by Elvert Barnes, Flickr) **Solution** 1. Clearly, a bowling ball is a  sphere , so spherical coordinates would probably  work  best here. The origin should be located at the physical center of the ball. There is no obvious choice for how the x𝑥-, y𝑦- and z𝑧-axes should be oriented. Bowling balls normally have a weight block in the center. One possible choice is to align the z𝑧-axis with the axis of symmetry of the weight block. 2. A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. The z𝑧-axis should probably point upward. The x𝑥- and y𝑦-axes could be aligned to point east and north, respectively. The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port. 3. A cone has several kinds of symmetry. In cylindrical coordinates, a cone can be represented by equation z=kr,𝑧=𝑘𝑟, where k𝑘 is a constant. In spherical coordinates, we have seen that surfaces of the form φ=cφ=𝑐 are half-cones. Last, in rectangular coordinates, elliptic cones are  quadric surfaces  and can be represented by equations of the form z2=x2a2+y2b2.𝑧2=𝑥2𝑎2+𝑦2𝑏2. In this case, we could choose any of the three. However, the equation for the  surface  is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a  component  that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. Choose the z𝑧-axis to align with the axis of the cone. The  orientation  of the other two axes is arbitrary. The origin should be the bottom point of the cone. 4. A pipeline is a  cylinder , so cylindrical coordinates would be best the best choice. In this case, however, we would likely choose to orient our z𝑧-axis with the center axis of the pipeline. The x𝑥-axis could be chosen to point straight downward or to some other logical direction. The origin should be chosen based on the problem statement. Note that this puts the _z𝑧_-axis in a horizontal  orientation , which is a little different from what we usually do. It may make sense to choose an unusual  orientation  for the axes if it makes sense for the problem. 5. A football has rotational symmetry about a central axis, so cylindrical coordinates would  work  best. The z𝑧-axis should align with the axis of the ball. The origin could be the center of the ball or perhaps one of the ends. The position of the x𝑥-axis is arbitrary. ##### Exercise 12.7.712.7.7 Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?  How should we orient the coordinate axes? **Hint** **Answer** ## Key Concepts - In the  cylindrical coordinate system , a point in space is represented by the ordered triple (r,θ,z),(𝑟,θ,𝑧), where (r,θ)(𝑟,θ) represents the polar coordinates of the point’s projection in the xy𝑥𝑦-plane and _z_ represents the point’s projection onto the z𝑧-axis. - To convert a point from cylindrical coordinates to Cartesian coordinates, use equations x=rcosθ,y=rsinθ,𝑥=𝑟cos⁡θ,𝑦=𝑟sin⁡θ, and z=z.𝑧=𝑧. - To convert a point from Cartesian coordinates to cylindrical coordinates, use equations r2=x2+y2,tanθ=yx,𝑟2=𝑥2+𝑦2,tan⁡θ=𝑦𝑥, and z=z.𝑧=𝑧. - In the  spherical coordinate system , a point P𝑃 in space is represented by the ordered triple (ρ,θ,φ)(ρ,θ,φ), where ρρ is the distance between P𝑃 and the origin (ρ≠0),θ(ρ≠0),θ is the same angle used to describe the location in cylindrical coordinates, and φφ is the angle formed by the positive z𝑧-axis and line segment OP¯¯¯¯¯¯¯¯𝑂𝑃¯, where O𝑂 is the origin and 0≤φ≤π.0≤φ≤π. - To convert a point from spherical coordinates to Cartesian coordinates, use equations x=ρsinφcosθ,y=ρsinφsinθ,𝑥=ρsin⁡φcos⁡θ,𝑦=ρsin⁡φsin⁡θ, and z=ρcosφ.𝑧=ρcos⁡φ. - To convert a point from Cartesian coordinates to spherical coordinates, use equations ρ2=x2+y2+z2,tanθ=yx,ρ2=𝑥2+𝑦2+𝑧2,tan⁡θ=𝑦𝑥, and φ=arccos(zx2+y2+z2−−−−−−−−−−√)φ=arccos⁡(𝑧𝑥2+𝑦2+𝑧2). - To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,θ=θ,𝑟=ρsin⁡φ,θ=θ, and z=ρcosφ.𝑧=ρcos⁡φ. - To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=r2+z2−−−−−−√,θ=θ,ρ=𝑟2+𝑧2,θ=θ, and φ=arccos(zr2+z2−−−−−−√).φ=arccos⁡(𝑧𝑟2+𝑧2). ## Glossary ** cylindrical coordinate system ** ** spherical coordinate system ** ## Contributors and Attributions - Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. 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