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From: majka@ubc-vision.UUCP (Marc Majka)
Newsgroups: net.graphics
Subject: re: parametric equations
Message-ID: <127@ubc-vision.UUCP>
Date: Mon, 28-Apr-86 14:06:49 EDT
Article-I.D.: ubc-visi.127
Posted: Mon Apr 28 14:06:49 1986
Date-Received: Mon, 28-Apr-86 17:08:20 EDT
Organization: UBC Computational Vision Lab, Vancouver, B.C., Canada
Lines: 36


The problem of finding parametric equations for a surface relies upon being
able to parameterize the surface.  A parameter, or more generally a set of
parameters form a new coordinate system in which you hope that the
description of the surface will be simple.  It is very difficult to find
such a coordinate system for a general surface.  For a curve (or a section
of a curve) the solution is easy: the parameter has an initial value at an
end point of the curve, and a unique path along which its value changes.  

Trying to extend this idea to surfaces lands one immediatly in deep water.
Where do the parameters have zero value (i.e. the origin of the new
coordinate system), and which way do they go?  The choices get very
arbitrary.  There are "local" solutions which work well.  By "local", I mean
that for any point on the surface, a coordinate system exists with that
point at the origin.  A good example is Riemannian normal coordinates (RNC),
which essentially are "polar" coordinates around that origin, with
"directions" and "lengths" based on the geodesics (shortest-path curves) at
that point.

Trouble is, there just are no good natural coordinate systems like RNC with
a global characterization.  You may be able to construct a principled way
(read algorithm) of making some arbitrary choices for some domain of
intrest, but you may end up with artifacts and secondary problems that are
painful.  Even then, you still will not be sure that your new coordinate
system will afford a "good" parameterization of the surface.

Grains of salt to be taken with this posting:

1) I am talking about a very general notion of surfaces. You may find easy
solutions for special cases.

2) For those of you with a background in Differential Geometry: please
excuse my cavalier use of terminology.  I am trying to make this simple.

---
Marc Majka