# Coordinate Systems

Give me a firm place to stand and i will give your rounding errors far away

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``