1.3.4. Spherical Coordinates

Table of Contents

Idea of spherical coordinates

Spherical coordinates are used to describe positions in three dimensional space when distances from a central point and directions matter more than rectangular distances along the $x$, $y$, and $z$ axes. They are especially useful for spheres, radial fields, and problems with symmetry around a point.

In spherical coordinates, a point is specified by three numbers, usually written as $(r,\theta,\phi)$ or sometimes $(r,\phi,\theta)$ depending on convention. Because conventions differ between books, it is important to define the angles clearly every time.

A common physics convention is this:

\[
(r,\theta,\phi)
\]

where $r$ is the distance from the origin, $\theta$ is the polar angle measured downward from the positive $z$ axis, and $\phi$ is the azimuthal angle measured in the $xy$ plane from the positive $x$ axis.

Meaning of the three coordinates

The coordinate $r$ tells how far the point is from the origin. It is always nonnegative:

\[
r \ge 0
\]

The angle $\theta$ tells how far the direction tilts away from the positive $z$ axis. Its range is usually

\[
0 \le \theta \le \pi
\]

The angle $\phi$ tells the direction of rotation around the $z$ axis in the horizontal plane. Its range is usually

\[
0 \le \phi < 2\pi
\]

A point near the positive $z$ axis has small $\theta$. A point in the $xy$ plane has $\theta = \frac{\pi}{2}$. A point on the negative $z$ axis has $\theta = \pi$.

In the common physics convention for spherical coordinates,
\[
r = \text{distance from the origin}
\]
\[
\theta = \text{angle from the positive } z \text{ axis}
\]
\[
\phi = \text{angle in the } xy \text{ plane from the positive } x \text{ axis}
\]
Always check the convention being used, because some texts swap $\theta$ and $\phi$.

Visual picture

You can imagine locating a point in three steps. First, draw a line from the origin to the point. The length of this line is $r$. Next, measure the tilt of that line away from the positive $z$ axis, which gives $\theta$. Finally, look at the shadow of the line in the $xy$ plane and measure its direction from the positive $x$ axis, which gives $\phi$.

Rendering scientific sketch…

Spherical coordinates

Relation to Cartesian coordinates

Spherical coordinates can be converted into Cartesian coordinates using the geometric meaning of the angles.

The projection of the point onto the $xy$ plane has length

\[
\rho = r\sin\theta
\]

Then in the plane,

\[
x = \rho\cos\phi, \qquad y = \rho\sin\phi
\]

So the full relations are

\[
x = r\sin\theta\cos\phi
\]

\[
y = r\sin\theta\sin\phi
\]

\[
z = r\cos\theta
\]

These formulas show how the radial distance and the two angles determine the point in ordinary rectangular coordinates.

For the standard physics convention,
\[
x = r\sin\theta\cos\phi
\]
\[
y = r\sin\theta\sin\phi
\]
\[
z = r\cos\theta
\]
These are the key conversion formulas from spherical to Cartesian coordinates.

From Cartesian to spherical

If the Cartesian coordinates $(x,y,z)$ are known, then spherical coordinates can be found from:

\[
r = \sqrt{x^2 + y^2 + z^2}
\]

The polar angle satisfies

\[
\cos\theta = \frac{z}{r}
\]

\[
\theta = \cos^{-1}\left(\frac{z}{r}\right)
\]

The azimuthal angle satisfies

\[
\tan\phi = \frac{y}{x}
\]

so in practice one must choose the correct quadrant for $\phi$. In many applications, a two argument arctangent function is used.

Coordinate surfaces

Each spherical coordinate can be held constant while the others vary. This produces useful geometric surfaces.

If $r$ is constant, the surface is a sphere centered at the origin.

If $\theta$ is constant, the surface is a cone with vertex at the origin and axis along the $z$ axis.

If $\phi$ is constant, the surface is a vertical half plane passing through the $z$ axis.

These surfaces help explain why spherical coordinates are natural for centrally symmetric problems.

Constant coordinate	Surface produced
$r = \text{constant}$	Sphere
$\theta = \text{constant}$	Cone
$\phi = \text{constant}$	Half plane through $z$ axis

Special cases

Some values have simple meanings.

If $\theta = 0$, the point lies somewhere on the positive $z$ axis.

If $\theta = \pi$, the point lies somewhere on the negative $z$ axis.

If $\theta = \frac{\pi}{2}$, the point lies in the $xy$ plane.

If $r = 0$, the point is at the origin. At the origin, the angles do not matter, because every direction starts there.

Why spherical coordinates are useful

Spherical coordinates are often the best choice when the physics depends mainly on distance from a central point. This happens in problems involving spheres, planets, atoms, and radial fields. In such cases, equations and descriptions become simpler because the geometry matches the symmetry of the situation.

For example, a spherical surface of radius $R$ is described very simply as

\[
r = R
\]

In Cartesian coordinates, the same surface is written as

\[
x^2 + y^2 + z^2 = R^2
\]

The spherical form is much more direct.

Comparison with other systems

Spherical coordinates are for three dimensional space with symmetry about a point. They are related to polar coordinates, which describe points in a plane using a distance and one angle. In fact, spherical coordinates can be seen as a three dimensional extension of polar coordinates.

System	Coordinates	Best for
Cartesian	$(x,y,z)$	Box shaped geometry, straight axis descriptions
Cylindrical	$(\rho,\phi,z)$	Symmetry around an axis
Spherical	$(r,\theta,\phi)$	Symmetry around a point

Example

Suppose a point has spherical coordinates

\[
(r,\theta,\phi) = \left(4,\frac{\pi}{3},\frac{\pi}{6}\right)
\]

Then

\[
x = 4\sin\frac{\pi}{3}\cos\frac{\pi}{6}
\]

\[
y = 4\sin\frac{\pi}{3}\sin\frac{\pi}{6}
\]

\[
z = 4\cos\frac{\pi}{3}
\]

Using

\[
\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, \qquad \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \qquad \sin\frac{\pi}{6} = \frac{1}{2}, \qquad \cos\frac{\pi}{3} = \frac{1}{2}
\]

we get

\[
x = 4 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = 3
\]

\[
y = 4 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \sqrt{3}
\]

\[
z = 4 \cdot \frac{1}{2} = 2
\]

So the Cartesian coordinates are

\[
(x,y,z) = (3,\sqrt{3},2)
\]

Final remarks

Spherical coordinates describe a point by its distance from the origin and two angles. Their strength comes from matching the geometry of spheres and radial symmetry.

The essential facts about spherical coordinates are:
\[
r \ge 0, \qquad 0 \le \theta \le \pi, \qquad 0 \le \phi < 2\pi
\]
\[
x = r\sin\theta\cos\phi, \qquad y = r\sin\theta\sin\phi, \qquad z = r\cos\theta
\]
Constant $r$ gives spheres, constant $\theta$ gives cones, and constant $\phi$ gives half planes.

Once these ideas are clear, spherical coordinates become a very natural way to describe three dimensional space.

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