Summary

When working on problems in three dimensional space it is often required convert between two or more co-ordinate systems. This article presents the formulae to convert between Cartesian and Spherical co-ordinate systems.

Definitions

rr:Radial distance
φ\varphi:Zenith angle ( 0≤φ≤π0 \le \varphi \le \pi )
θ\theta:Azimuth angle ( 0≤θ≤2π0 \le \theta \le 2\pi )

Introduction

A common procedure when operating on 3D objects is the conversion between spherical and Cartesian co-ordinate systems. This is a rather simple operation however it often results in some confusion.

The spherical coordinates system defines a point in 3D space using three parameters, which may be described as follows:

  1. The radial distance from the origin (O) to the point (P), r
  2. The zenith angle, φ\varphi between the zenith reference direction (z-axis) and the line OP with 0≤φ≤π0 \leq \varphi \leq \pi
  3. The azimuth angle, θ\theta between the azimuth reference direction (x-axis) and the orthogonal projection of the line OP of the reference plane (x-y plane) with 0≤θ≤2π0 \leq \theta \leq 2\pi

Conversion from Cartesian to Spherical

Conversion Formula

We may convert a given a point in Cartesian co-ordinates (x,y,z) to spherical co-ordinates using the following formulas:

r=(x2+y2+z2)φ=arccos⁡(y/r)θ=arctan⁡(y/x) \begin{aligned} r &= \sqrt{ (x^2 + y^2 + z^2) } \\ \\ \varphi &= \arccos\left(y/r\right) \\ \\ \theta &= \arctan\left(y/x\right) \end{aligned}

Notes on Computational Implementations

One must take care when implementing this conversion as using a standard atan function will only yield the correct spherical coordinates if the point is in the first or fourth quadrant (positive x values).

Instead the function atan2 should be used which takes the coordinates x and y as parameters and returns atan(y/x) taking into account the quadrant x and y lie in by adjusting the values as follows:

atan2(y,x)={arctan(y/x)x>0π+arctan(y/x)y≤0,x<0−π+arctan(y/x)y<0,x<0π/2y>0,x=0−π/2y<0,x=0undefinedy=0,x=0 atan2(y,x) = \left\{ \begin{array}{l l} arctan(y/x) & \quad x \gt 0\\ \\ \pi + arctan(y/x) & \quad y \leq 0, x \lt 0\\ \\ -\pi +arctan(y/x) & \quad y \lt 0, x \lt 0 \\ \\ \pi/2 & \quad y \gt 0, x = 0 \\ \\ -\pi/2 & \quad y \lt 0, x = 0 \\ \\ \text{undefined} & \quad y = 0, x = 0 \end{array} \right.

However take note of the parameter order of this function on your platform as it can differ. For example in C++ the interface is atan2(y,x) while in excel it is atan2(x,y).

Conversion from Spherical to Cartesian

Conversion Formula

Conversely spherical coordinates may be converted to Cartesian coordinates using the following formulas:

x=rsin⁡(φ)cos⁡(θ)y=rsin⁡(φ)sin⁡(θ)z=rcos⁡(φ) \begin{aligned} x &= r\sin\left(\varphi\right) \cos\left(\theta\right) \\ \\ y &= r\sin\left(\varphi\right) \sin\left(\theta\right) \\ \\ z &= r\cos\left(\varphi\right) \end{aligned}