Spherical Coordinates

Main idea

Spherical coordinates locate a point the way a globe does: how far it is from the center ( $\rho$ ), its compass direction ( $\theta$ ), and how far down it is from the north pole ( $\varphi$ ). Spheres and cones turn into one-line equations.

x = {\htmlData{sym=0}{\rho}}\sin{\htmlData{sym=1}{\varphi}}\cos{\htmlData{sym=2}{\theta}}, \quad y = {\htmlData{sym=0}{\rho}}\sin{\htmlData{sym=1}{\varphi}}\sin{\htmlData{sym=2}{\theta}}, \quad z = {\htmlData{sym=0}{\rho}}\cos{\htmlData{sym=1}{\varphi}}, \qquad {\htmlData{sym=0}{\rho}}^2 = x^2 + y^2 + z^2

What the symbols mean

$\rho$
Distance from the origin
Always at least 0.
$\varphi$
Angle from the north pole
From 0 (straight up) to π (straight down).
$\theta$
Longitude
The same angle as in cylindrical coordinates.

Explanation

Spherical coordinates are simply the globe's way of giving an address: how far you are from the center, which way around (like longitude), and how far down from the north pole (like latitude, but counted from the top). They're named after the sphere because "stay a fixed distance from the center" draws one. The formulas for $x$ , $y$ and $z$ look like a pile of sines and cosines, but they're built from three plain pieces: $\rho$ (distance from the origin), $\theta$ (the compass direction, the same angle as in cylindrical coordinates) and $\varphi$ (the tilt down from straight up). Bear with it; we'll see where each sine and cosine comes from below.

Cylindrical coordinates were made for things that are round around an axis. Spherical coordinates are made for things that are round around a point: a ball, the Earth, the signal spreading from an antenna. In $x$ , $y$ , $z$ a sphere is $x^2 + y^2 + z^2 = 9$ ; here it's just $\rho = 3$ .

Simple shapes:

$\rho = c$ : a sphere of radius $c$ .
$\varphi = c$ : a cone around the $z$ -axis, since every point is tilted the same angle from straight up. The special case $\varphi = \frac\pi2$ is tilted all the way to horizontal, which is the $xy$ -plane.
$\theta = c$ : a half-plane, the same as in cylindrical coordinates.

Converting. Here's where the formulas come from. Picture the arrow of length $\rho$ from the origin, tilted $\varphi$ from vertical. Its height is the side next to the angle, $z = \rho\cos\varphi$ . Its shadow on the floor is the side opposite the angle, $\rho\sin\varphi$ , and that shadow's length is exactly the distance from the $z$ -axis, the cylindrical $r$ . Now apply polar coordinates to the shadow: $x = r\cos\theta$ and $y = r\sin\theta$ with $r = \rho\sin\varphi$ . That's all three formulas.

A common mistake. It's tempting to measure $\varphi$ up from the equator, the way latitude works on a map. It isn't measured that way: $\varphi$ starts at $0$ straight up at the north pole and runs to $\pi$ straight down. So latitude is $90° - \varphi$ , and the equator is $\varphi = \frac\pi2$ , not $0$ .

See it

A point on a sphere of radius 2 with its compass direction fixed at θ = 45°, while φ grows. As φ goes from near 0 to near π, the point slides from the north pole down a line of longitude to the south pole, crossing the equator at φ = π/2 (90°). That crossing is why latitude is 90° − φ.

Worked examples

Example 1

Convert $\left(\rho, \theta, \varphi\right) = \left(2, \frac\pi4, \frac\pi3\right)$ to Cartesian.

1
First find the shadow on the floor, the distance from the $z$ -axis: $\rho\sin\varphi = 2\cdot\frac{\sqrt3}{2} = \sqrt3$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find $\rho$ for the point $(1, 2, 2)$ .

Problem 2+5 XP

Convert $(\rho, \theta, \varphi) = \left(4, 0, \frac\pi2\right)$ to Cartesian.

Problem 3+5 XP

Find $\varphi$ for the point $(0, 0, -5)$ .

Problem 4+5 XP

What does $\varphi = \frac\pi4$ describe?

Problem 5+5 XP

What does $\rho = 2\cos\varphi$ describe?

Problem 6+5 XP

On a spherical Earth of radius $6371$ km, a place at latitude $30°$ N has polar angle $\varphi = 60°$ , measured down from the North Pole. How far is it above the plane of the equator, $z = \rho\cos\varphi$ ?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Convert $(x, y, z) = (1, 1, \sqrt6)$ to spherical coordinates $(\rho, \theta, \varphi)$ .

Where it's used

Antenna radiation patterns, planetary gravity and atomic orbitals are all written in spherical coordinates, and so is every latitude/longitude on Earth.

Cylindrical Coordinates Limits of Functions of Several Variables