Coordinate systems in $\mathbb{R}^3$

back to contents

Definition: Cylindrical coordinates

In the cylindrical coordinate system, a point $P\in\mathbb{R}^3$ is represented by the ordered triple $(r, \theta, z)$ where

• $(r,\theta)$ are the polar coordiantes of the point’s projection in the $xy$-plane
• $z$ is the usual $z$ Cartesian coordinate

Coordinate conversions

$$\begin{array}{lcl} (r,\theta,z)\to (x,y,z) & \qquad & (x,y,z) \to (r,\theta,z) \\\\x = r\cos \theta & & r^2 = x^2+y^2 \\y = r\sin\theta & & \displaystyle \tan\theta = \frac{y}{x} \qquad \theta\in[0,2\pi] \\z = z & & z= z \end{array} $$

Definition: Spherical coordinates

In the spherical coordinate system, a point $P\in\mathbb{R}^3$ is represented by the ordered triple $(\rho,\theta,\phi)$ where

• $\rho$ is the distance between $P$ and the origin $O$
• $\theta$ is the same as cylindrical coordinates
• $\phi\in[0,\pi]$ is the angle formed between positive $z$ axis $\vec{e}_3$ and vector $\vec{OP}$

Coordinate conversions

$$\begin{array}{lcl} (\rho,\theta,\phi)\to (x,y,z) & \qquad & (x,y,z) \to (\rho,\theta,\phi) \\\\x = \rho \sin \phi \cos \theta & & {\rho}^2 = x^2+y^2 + z^2 \\y = \rho \sin \phi \sin \theta & & \displaystyle \tan\theta = \frac{y}{x} \qquad \theta\in[0,2\pi] \\z = \rho \cos \phi & & \displaystyle\phi= \arccos \left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right) \end{array} $$