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Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude. [1] It is the simplest, oldest and most widely used type of the various spatial reference systems that are in use, and forms the basis for most others.
For a geographic coordinate system of the Earth, the fundamental plane is the Equator. Astronomical coordinate systems have varying fundamental planes: [2] The horizontal coordinate system uses the observer's horizon. The Besselian coordinate system uses Earth's terminator (day/night boundary). [3] This is a Cartesian coordinate system (x, y, z).
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). [9]
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them.
Calculating the distance between geographical coordinates is based on some level of abstraction; it does not provide an exact distance, which is unattainable if one attempted to account for every irregularity in the surface of the Earth. [1] Common abstractions for the surface between two geographic points are: Flat surface; Spherical surface;
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.