Associated Topics || Dr. Math Home || Search Dr. Math

### Circles on the Surface of a Sphere

```
Date: 08/11/99 at 12:32:56
From: Mark Shen
Subject: Circles on the Surface of a Unit Sphere and their
Transformation from Cartesian to Spherical Coordinates

The Problem:

A circle in the (X,Y,Z) coordinate system has the equation:

x^2 + z^2 = r^2   and   y = k   ..........................(Eq. 1)

Here, 'r' is the radius and 'k' is a constant. How do I convert this
equation/relation that is in terms of X, Y, and Z into a function that
is defined in terms of theta and phi, spherical coordinates? I want to
be able to make a 2-D graph (with axes theta and phi) of this new
relation. The spherical coordinate rho is not taken into consideration
in this 2-D graph.

I believe that when converted into theta and phi coordinates, the 2-D
graph will be a circle. I can show it is numerically - that is, I can
convert individual points on the (X,Y,Z) circle into theta and phi
coordinates. The result was a circle on the theta and phi coordinates.
The specific circle equation used was:

x^2 + z^2 = 0.25   and   y = sqrt(0.75)   ................(Eq. 2)

Obviously, 0.5 is the radius and y = sqrt(0.75) is the 2-D plane in
which the circle exists. The numbers were picked for convenience (but
not at first glance) so that rho (in the spherical system) would equal
one. Using the conversions below, I was able to generate a circle on
theta and phi axes.

x = sin(phi)*cos(theta)     rho = 1
y = sin(phi)*sin(theta)
z = cos(phi)

How can the equation below (Eq. 3, which is Eq. 2 in theta and phi) be
converted to a circle equation?

(sin(phi)*cos(theta))^2 + (cos(phi))^2 = 0.25   ..........(Eq. 3)

=======================================================

This problem is a small part of a larger problem: On a unit sphere in
the Cartesian coordinate system, you project a cone, with half-angle
gamma and apex at the origin, onto the surface of the sphere. The
resulting intersection of these two objects is a circle. If you
convert this circle, which is in the Cartesian system, into spherical
coordinates - theta and phi, is the resulting graph a circle as well?
How do you prove that this is true, with the exception of a few
special combinations of theta and phi? At first glance, it seemed easy
to prove; and the proof almost seems to be in the 'definition' of the
spherical coordinate system. However, I must be too dense to see it.

Please help. Thanks.

Mark Shen
```

```
Date: 08/11/99 at 15:20:59
From: Doctor Rob
Subject: Re: Circles on the Surface of a Unit Sphere and their
Transformation from Cartesian to Spherical Coordinates

Thanks for writing to Ask Dr. Math!

Probably the simplest way to see that the curve is a circle is to use
cylindrical coordinates instead of spherical ones:

x = R*cos(theta)
y = Z
z = R*sin(theta)

In your situation, the equations become

Z = sqrt(0.75)
R = sqrt(0.25) = 0.5

The equation R = 0.5 is that of a right circular cylinder with radius
0.5 and axis the Z-axis. The equation Z = constant is the equation of
a plane perpendicular to the Z-axis. Their intersection is a circle,
which is the circle you were seeking.

If you must use spherical coordinates, you have to use the additional
equation:

y = sqrt(0.75),

or

sin(phi)*sin(theta) = sqrt(0.75).

Using this to graph the phi-versus-theta relation should reveal the
graph of a circle.

By the way, if you square this last equation and use the identity
sin^2(u) + cos^2(u) = 1, you can derive your Eq. 3 above.

You would be even better advised to choose the following coordinate
transformation instead of the one you used:

x = sin(phi)*cos(theta)
y = cos(phi)
z = sin(phi)*sin(theta)

Then your equations will reduce to:

rho = 1
sin^2(phi) = 0.25
cos(phi) = sqrt(0.75)
phi = Pi/6

This is a cone with vertex at the origin and gamma = Pi/6, intersected
with the sphere rho = 1. Here theta can take any value at all.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
College Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/