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 3dimensional 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 3D counterparts in spherical form r =
f(theta, phi).
In each case, Graphing Calculator is "tricked" into
graphing an equation in 3D 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 3dimensional 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 2D graph. Click the picture to open
Graphing Calculator. 
This sphere was what I expected for the graph to the
left, too, based on the 2D graph. Did you? 
What about some cardioids?


This looks very much like one of the figures above,
but take a closer look. 
This is what I think of as a 3D 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. 
What about the roses?
This 2D 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 xyplane? 
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 3D graphs were a great visualization exercise for me.
I hope you enjoyed them! PLAY WITH YOUR MATH!
Return to D Hembree's EMAT
6680 page
Emailto : dhembree@coe.uga.edu