Generating Polytope Intersection Configurations from a Symbolic Description Using CLP(ℜ)

Abstract

Given two polytopes whose coordinates are not known, and a list Iof pairs of edges from the two polytopes which intersect, we wish to place the vertices on the plane (i.e., assign coordinates to the vertices) in a way which is consistent with I. Elements of Idistinguish between intersections that occur at vertices or at interior points of edges (i.e., points which are not vertices), but do not give either the absolute or relative position of an intersection. In addition to being consistent with I, a placement should satisfy certain aesthetic rules; e.g., a placement should make the polytopes appear regular rather than oddly-shaped when possible. This problem is treated as a constraint satisfaction problem. A set of arithmetic constraints on the coordinates of the vertices as well as certain auxilliary variables is generated. A search method is then used to find a solution to these constraints. This system is implemented in CLP(ℜ) and makes use of CLP(ℜ)’ s built-in constraint solution facilities where possible.