Complex Shapes

Date: Thu, 3 Nov 94 11:23:15 EST
From: Brian Sledd
Subject: Math Question

Dr. Math,

Please describe how to come up with mathematical "descriptions" 
for complex shapes (torus, witches hat, etc.) using three-
dimensional math.  

Brian

Date: Thu, 3 Nov 1994 20:46:52 -0500 (EST)
From: Dr. Ethan
Subject: Re: Math Question

Hi Brian,

Essentially you can do this the same way you mathematically 
explain things like parabolas in the plane; you just have to 
use two variables to parameterize it (that is the name for a 
mathematical description). Do you have any means for graphing 
three-dimensional objects? Like Mathematica (I'm sure there are 
others but I don't know of any). It would make this a lot 
easier if I could just send you some parameterizations to try.

But for starters:

A Sphere of radius one is all (x,y,z) that satisfy this 
equation:

     x^2 + y^2 + z^2 = 1

A Torus is all (x,y,z) that satisfy:

     z^2 = r^2 -(Sqrt[x^2 + y^2] - a)^2

There are a dozen ways to parameterize these and similar 
objects. If you have access to a graphing program tell me 
and I will try to send you some parameterizations that you can 
use.

-Doctor Ethan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/