The Spherical Coordinate System

# The Spherical Coordinate System

Apart from the Cartesian coordinate system and The Cylindrical Coordinate System, there is also another important coordinate system known as the spherical coordinate system to represents points in $\mathbb{R}^3$. Every point $P(x, y, z)$ can be represented as the ordered triple $(r, \theta, \phi)$.

• $r$ represents the distance from $P$ to the origin $O$. It can also be seen as the length of the line segment $OP$. Often times "$\rho$" is used in place of $r$.
• $\theta$ represents the angle between the positive $x$-axis and the projection of $OP$ onto the $xy$-plane.
• $\phi$ represents the angle between the $z$-axis and the line segment $OP$

A point $(r, \theta, \phi)$ in spherical coordinates can be converted to Cartesian coordinates $(x, y, z)$ with the following formulas:

(1)
Graph Description
Equations in the form of $r = c$ where $c \in \mathbb{R}$ represents spheres centered at the origin and with radius $r$. This should make sense as $r^2 = x^2 + y^2 + z^2$.
Equations in the form of $\theta = c$ where $c$ is a fixed angle represents half-planes.
Equations in the form of $\phi = c$ where $c$ is a fixed angle represents half circular cones.

## Example 1

Convert the spherical coordinates $\left (3, \pi, \frac{\pi}{3} \right )$ into Cartesian coordinates.

Using the formula for the $x$-coordinate and we get that:

(2)
\begin{align} \quad x = r \sin \phi \cos \theta = 3 \sin \left ( \frac{\pi}{3} \right ) \cos (\pi) = 3 \left ( \frac{\sqrt{3}}{2} \right ) (-1) = -\frac{3\sqrt{3}}{2} \end{align}

Using the formula for the $y$-coordinate and we get that:

(3)
\begin{align} \quad y = r \sin \phi \sin \theta = 3 \sin \left ( \frac{\pi}{3} \right ) \sin (\pi) = 3 \left ( \frac{\sqrt{3}}{2} \right ) (0) = 0 \end{align}

Using the formula for the $z$-coordinate and we get that:

(4)
\begin{align} \quad z = r \cos \phi = 3 \cos \left ( \frac{\pi}{3} \right ) = 3 \left ( \frac{1}{2} \right ) = \frac{3}{2} \end{align}

Therefore the spherical coordinates are $\left ( \frac{-3\sqrt{3}}{2}, 0, \frac{3}{2} \right )$.