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From: little@ubc-vision.CDN (Jim Little)
Newsgroups: net.math
Subject: Geometric Duality
Message-ID: <457@ubc-vision.CDN>
Date: Wed, 4-Jul-84 16:43:32 EDT
Article-I.D.: ubc-visi.457
Posted: Wed Jul  4 16:43:32 1984
Date-Received: Sat, 7-Jul-84 00:35:29 EDT
Organization: UBC Vision, Vancouver, B.C., Canada
Lines: 43


Duality is well defined for irregular polyhedra. For a good discussion,
see "Convex Polytopes" by Branko Gruenbaum, John Wiley & Sons, Ltd., 1967,
pp. 46-48.  The text is available from their London office.  A polytope is
a bounded convex polyhedron. 

A polytope can be defined as the intersection of a set of halfspaces; 
each half space is given by the equation:

                     Ax + By + Cz <= 1

For a polytope composed of n planes (the primal polytope), containing the 
origin in its interior, one can construct its geometric dual as follows:

1) each plane with equation

                     Ai x + Bi y + Ci z = 1

is transformed into a point whose coordinates are (Ai,Bi,Ci).

2) each point (s,t,u) is taken into the plane with equation

                     s x + t y + u z = 1

This transform takes planes into points and takes points into planes.  
Consider a vertex p=(s,t,u) of the primal polytope, 
lying on a set of faces of the primal.  For each face j it is the case that

                     Aj s + Bj t + Cj u = 1

From this it is clear that in the dual the plane corresponding to the vertex
(s,t,u) contains the point (Aj, Bj, Cj) if and only if the plane (Aj, Bj, Cj)
contains the vertex in the primal.

The condition that the origin does not lie on any of the faces of the
primal polytope is necessary so that the plane equations are well defined.
This transformation can be extended in a straightforward fashion to 
d>3 dimensions; the discussion is in terms of R3 only for clarity.

                                        Jim Little
                                        ubc-vision
                                        University of British Columbia