Subject: prolate / oblate spheroidal coordinates

From: Gustavo Mercier <[email-masked]>
Date: Wed, 24 Nov 1993 13:15:40 -0500 (EST)

Hi, Netters!

Although the question below may seem a simple one that should be answered 
with a trip to the library, it proved more difficult than expected!

I am writing a program to do some electrostatic computations and association
dynamics. To take advantage of symmetry I need to use prolate and oblate 
spheroidal coordinates. Unfortunately, I don't have the formulae for 
conversion between cartesian and these coordinate systems. More important 
apparently there are several definitions for these coordinate systems.

I am following papers on the solution of Laplace equation with ellipsoid 
symmetry as described by Beveridge in the late '70's. They give a definition
for prolate spheroidal coordinate as follows

	lambda = (|r-ra| + |r-rb|) / 2d
	mu = (|r-ra| - |r-rb|) / 2d
	phi = angle of rotation around the major axis.

r = point in space;
ra = focus 1
rb = focus 2
2d = distance between foci

These coordinates have the following boundaries:

	0 <= lambda <= infinity
	-1 <= mu <= 1
	0 <= phi < 2 Pi

The conversion of these to oblate spheroidal is not clear to me.
Can anybody help?

Note that this definition of prolate spheroidal coordinates does not follow
other conventions. Specifically, the Package VectorAnalysis.m in Mathematica
follows a very different convention and formulae!

gus mercier
[email-masked]