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From: jbn@glacier.ARPA (John B. Nagle)
Newsgroups: comp.graphics
Subject: Tough geometry problem
Message-ID: <13111@glacier.ARPA>
Date: Sun, 23-Nov-86 22:16:22 EST
Article-I.D.: glacier.13111
Posted: Sun Nov 23 22:16:22 1986
Date-Received: Mon, 24-Nov-86 21:39:46 EST
Organization: Stanford University, IC Laboratory
Lines: 40
Keywords: superquartics, computational geometry, vector calculus


     I need to calculate the distance between two objects described using
the superquartic primitives proposed by Pentland (in SRI Tech Note 357).
Anybody tackle this one yet?  I forsee about a month of work ahead, although
maybe I can use Macsyma to do some of the grunt work.

     If you're interested in what this is about, read on.

     The idea is to have a clean way to approximately describe real-world
objects.  The usual polyhedral approach leads to a very verbose description
of objects as simple as cylinders and spheres, and the description of (say)
animals or plants as polyhedra is very painful.  An alternative approach is 
to build up object from simple volumetric primitives, such as cylinders,
cones, spheres, and the like.  GMSolid, General Motors' entry in the solid
modelling game, works in this way.

     Pentland has an extension to this.  Instead of limiting the objects to
quartics, (things describable with exponents no larger than 2), he uses
superquartics, which allow bigger exponents.  Imagine a display of a
sphere attached to a "squareness" control; as you turn up the "squareness",
the corners become more square.  An intermediate figure looks like a TV
screen, and higher values look like later model TV screens; eventually
one gets a cube.  

     That's the basic figure.  One is then allowed three distortions of
the superquartic; stretching, bending (in a circular sense only) and
tapering.  All of these can be modelled as simple distortions of the
metric.  The resulting distored primitives can then be combined, using
the obvious operations of union and difference (which latter operation
corresponds to making a "hole" in something) to make complex objects.

      Given all this, I want to be able to calculate whether two such
constructed objects collide or if not, how far apart they are at the
closest point.   There are two parts of the problem; first, solving it
for figures without holes, and then dealing with holes, which complicates
the problem considerably.



					John Nagle