Everything about Spherical Coordinates totally explained
In
mathematics, the
spherical coordinate system is a
coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the
zenith angle from the positive z-axis to the point, and the
azimuth angle from the positive x-axis to the
orthogonal projection of the point in the x-y plane.
Notation
Several different conventions exist for representing the three coordinates. In accordance with the International Organisation for Standardisation (
ISO 31-11), in physics they're typically notated as (
r,
θ,
φ) for radial distance, zenith, and azimuth, respectively.
In (American) mathematics, the notation for zenith and azimuth are reversed as
φ is used to denote the zenith angle and
θ is used to denote the azimuthal angle. A further complication is that some mathematics texts list the azimuth before the zenith, but this convention is
left-handed and should be avoided. The mathematical convention has the advantage of being most compatible in the meaning of
θ with the traditional notation for the two-dimensional
polar coordinate system and the three-dimensional
cylindrical coordinate system, while the "physics" convention has broader acceptance geographically. Some users of the "physics" convention also use
φ for polar coordinates to avoid the first problem (as is the standard ISO for
cylindrical coordinates). Other notation uses
ρ for radial distance. The notation convention of the author of any work pertaining to spherical coordinates should always be checked before using the formulas and equations of that author. This article uses the standard physics convention.
Definition
The three coordinates (
r,
θ,
φ) are defined as:
- r ≥ 0 is the distance from the origin to a given point P.
- 0 ≤ θ ≤ π is the angle between the positive z-axis and the line formed between the origin and P.
- 0 ≤ φ < 2π is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.
φ is referred to as the azimuth, while
θ is referred to as the zenith, colatitude or polar angle.
θ and
φ lose significance when
r = 0 and
φ loses significance when sin(
θ) = 0 (at
θ = 0 and
θ = π).
To plot a point from its spherical coordinates, go
r units from the origin along the positive z-axis, rotate
θ about the y-axis in the direction of the positive x-axis and rotate
φ about the z-axis in the direction of the positive y-axis.
Coordinate system conversions
As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.
Cartesian coordinate system
The three spherical coordinates are obtained from
Cartesian coordinates by:
» Further Information
Get more info on 'Spherical Coordinates'.
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