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
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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? |
What about some cardioids?
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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 |
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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.
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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 2-D graph is an active link to Graphing Calculator
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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.
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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.
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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!
Return to D Hembree's EMAT
6680 page
Emailto : dhembree@coe.uga.edu