 Spherical Coordinates and Graphs

by

D. Hembree

Graphs in polar coordinates are an interesting topic in high school analysis and calculus and some students (and teachers) come to have a feel for what the graph of a particular equation will look like. However, it's often a much different matter to visualize a 3-dimensional counterpart in spherical coordinates.

As a polar coordinate system does in the plane, a spherical coordinate system locates a point in space in terms of its relation to a central point. In space three coordinates are needed. The conventional notation is (r, theta, phi), defined as shown in the figure below. Following is a simple presentation: a common polar graph is shown along with one or more 3-D counterparts in spherical form r = f(theta, phi).
In each case, Graphing Calculator is "tricked" into graphing an equation in 3-D view by including a phi term with a coefficient of zero, and then a graph is produced by replacing theta with phi.
Throughout this page, clicking on any 3-dimensional graph will open Graphing Calculator for the Macintosh computer   It's worth getting a bigger look at this and thinking about how it relates to the 2-D graph. Click the picture to open Graphing Calculator. This sphere was what I expected for the graph to the left, too, based on the 2-D graph. Did you?   This looks very much like one of the figures above, but take a closer look. This is what I think of as a 3-D cardiod   Still the same? This is unexpected! Where's the "loop"? Scroll down for another set of views!!

Here are two views inside the top right figure.  By limiting the range of phi, we can "cut the top off the figure" and see the missing loop inside Just another view with different shading. This 2-D graph is an active link to Graphing Calculator  After watching this animation, I thought "well of course" THIS was a surprise. You have to open this and watch the animation!

Spirals A hyperbolic, or reciprocal spiral. Notice the range on theta. The line y = 1 is an asymptote.  Notice the change in range for theta and phi. Phi was changed in order to "cut the spiral" and look inside. Is there an "asymptote" for this graph? This just looks like a tall vial. Look at it in Graphing Calculator. Do you know why it dips below the x-y-plane? A logarithmic, or equiangular spiral. Notice the range on theta and the scale on the graph.  The range of theta was restricted in order to get a better view on the screen. Theta is still restricted here and it allows you to see inside the figure. It's really hard to get a good view of theis figure.

These 3-D graphs were a great visualization exercise for me. I hope you enjoyed them! PLAY WITH YOUR MATH!