Coordinate Systems

Ways to describe the position of a point space.

Cartesian

A point is expressed as P = xi + yi + zi, where i, j and k are unit vectors parallel to the axes of the coordinate system.

To Cylindrical

r = sqrt(x²+y²)

θ = arctan y/x

z = z

To Spherical

r = sqrt(x+y+z)

θ = arctan y/x

φ = arctan( sqrt(x+y+z) / z )

Cylindrical

A point is made up of r, θ and z. θ is the angle around the z-axis. r is the distance to the point from the z-axis and z is the height of the point above the x-y plane.

To Cartesian

x = r cos θ

y = r sin θ

z = z

To Spherical

r = sqrt(r²+z²)

θ = θ

φ = arctan r/z

Spherical

A point is made up of r, θ and φ. r is the distance from the origin to the point. θ is the angle around the z-axis and φ is the angle between the z-axis and line to the point from the origin.

To Cartesian

x = r cos θ sin φ

y = r sin θ sin φ

z = r cos φ

To Cylindrical

θ = θ

r = r sin φ

z = r cos φ

Reference3D Math Primer for Game Programmers (Coordinate Systems) - 2011