parametric equations

[Gary Letourneau]

I am beginning some work on graphing 2-d and 3-d curves and surfaces and
I have run across the following problem:

If given an equation of a 3-dimensional surface in the form

f(x, y, z) = ...

and a range of values for x, y, and z

x1 < x < x2
y1 < y < y2
z1 < z < z2

is there an algorith for determining the parametric equations for the
same surface

fx(t) = ...
fy(t) = ...
fz(t) = ...

where 0 <= t <= 1

In short, is there an algorithmic way of generating parametric equations
from nonparametric ones?

I would appreciate any pointers, ideas, program code, etc. that anyone
would post.
Thanks in advance,
Gary Letourneau
letourneau@nlm-mcs

[Ken Turkowski]

In article <14...@nlm-mcs.ARPA> ga...@nlm-mcs.ARPA (Gary Letourneau) writes:
> If given an equation of a 3-dimensional surface in the form
> f(x, y, z) = ...
> and a range of values for x, y, and z

> x1 < x < x2, y1 < y < y2, z1 < z < z2

> is there an algorith for determining the parametric equations for the
> same surface

> fx(t) = ... , fy(t) = ... , fz(t) = ... , 0 <= t <= 1

First, you'll never get a full surface from a univariate function; you need
two variables:

fx(u,v), fy(u,v), fz(u,v), 0 <= u <= 1, 0 <= v <= 1

Otherwise, you'll get a curve instead of a surface.
--
Ken Turkowski @ CIMLINC, Menlo Park, CA
UUCP: {amd,decwrl,hplabs,seismo}!turtlevax!ken
ARPA: turtlevax!k...@DECWRL.DEC.COM

[Assoc for Computing Machinery]

An excellent book that covers the mathematical representations of parametric
curves, surfaces, and solid plus their generation from non-parametric
objects is one entitled _Geometric Modeling_ by Michael E. Mortenson from
John Wiley & Sons (1985). You should find everything you need in there
plus more about the analytical properties of these objects, transformations,
and some information on solid modeling.

David E. Lee
UCLA ACM Chairman