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.
In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects.
Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid.
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.
A spherical coordinate system is a three-dimensional curvilinear coordinate system that can be used to describe a point using the radial distance, the polar angle, and the azimuthal angle.
Spherical coordinates are a system for locating points in three-dimensional space using three values: the distance from the origin (\rho ρ), the angle down from the positive z z-axis (\phi ϕ), and the angle of rotation around the z z-axis (\theta θ).
Spherical coordinate system, In geometry, a coordinate system in which any point in three-dimensional space is specified by its angle with respect to a polar axis and angle of rotation with respect to a prime meridian on a sphere of a given radius.
Instructions: The animation above illustrates the geometry of the spherical coordinate system, showing its coordinate curves, surfaces, and basis vectors (explained below).