Spherical coordinates

Spherical coordinates can take a little bit to get used to. The radius $\rho$ isn't difficult to understand. The angle $\theta$ isn't so bad either, since it corresponds to the $\theta$ of the familiar polar coordinates. But some students have trouble grasping what the angle $\phi$ is all about. To help you understand spherical coordinates better, I've created a few CVTs where you can click and drag the blue points along the sliders to change the parameters.

Recall that spherical coordinates are defined as indicated in the following figure, which illustrates the spherical coordinates of the point P.

The coordinate $\rho$ is the distance from P to the origin. If the point Q is the projection of P to the xy-plane, then $\theta$ is the angle between the positive x-axis and the line from the origin to Q. Lastly, $\phi$ is the angle from the positive z-axis to the line segment from the origin to P.

The first CVT allows you to see how the location of a point changes as you vary $\rho$, $\theta$, and $\phi$. Drag the blue points along sliders to change the values of $\rho$, $\theta$, and $\phi$. (Sorry, due to font problems in the demo, I had to write $\rho$ as p, $\theta$ as $\Theta$, and $\phi$ as $\varphi$.)

The purple point moves to reflect the corresponding position in Cartesian coordinates (i.e., it is P in the above figure). The green dot is the projection of the point in the xy-plane (i.e., it is Q in the above figure). The projection gives perspective and helps in visualizing $\theta$.

Notice how you can obtain any point even though we restrict $\rho$$\ge$ 0, 0$\le$$\theta$ < 2$\pi$, and 0$\le$$\phi$$\le$$\pi$. Can you see why we only need $\phi$ to go up to $\pi$?

These restrictions removed much of the non-uniqueness of spherical coordinates. Notice there is still non-uniqueness at $\rho$ = 0, at $\phi$ = 0 and at $\phi$ = $\pi$ (when any of these conditions are true, you can change the value of one or more of the other coordinates without moving the point).

Simple spherical coordinate surfaces

These three next CVTs may help you understand what each of three spherical coordinates means. They show what the surfaces $\phi$ = constant, $\theta$ = constant, and $\rho$ = constant look like. The value of the constant is determined by the position of the sliders. In all cases, we restrict the surfaces to the region $\rho$ < 5.

Constant $\phi$

If I tell you a certain point in spherical coordinates has $\phi$ = $\pi$/3, what does that mean? Take a look at what surfaces are defined by the equation $\phi$ = constant.

The surface $\phi$ = constant is simply a single cone, pointing either upward or downward. If you know that $\phi$ = $\pi$/3, then you know the point is somewhere on a (wide) single cone that opens upward, i.e., the equation $\phi$ = $\pi$/3 specifies a surface that is a single cone opening upward.

Constant $\theta$

The surface $\theta$ = constant is a half-plane off the z-axis. (It is plotted as a half-disk only because we restrict the plot to $\rho$ < 5.)

So, if a point has $\theta$ = $\pi$/4, then you know the point is on the half of the yz-plane where y values are positive. The equation $\theta$ = $\pi$/4 is the equation for this half-plane.

Constant $\rho$

I believe most people understand what $\rho$ = 3 means. It is the sphere of radius 3 centered at the origin. In general, the surface $\rho$ = constant is a sphere of radius $\rho$ centered at the origin.

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