Discrete Applied Mathematics
Checking the convexity of polytopes and the planarity of subdivision
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Checking geometric programs or verification of geometric structures
Selected papers from the 12th annual symposium on Computational Geometry
Nonregular Triangulations, View Graphs of Triangulations, and Linear Programming Duality
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
| Hi-index | 0.00 |
Guibas conjectured that given a convex polygon P in the xy-plane along with two triangulations of it, T1 and T2 that share no diagonals, it is always possible to assign height values to the vertices of P such that P ∪ T1 ∪ T2 becomes a convex 3-polytope. Dekster found a counter example but left open the questions of deciding if a given configuration corresponds to a convex 3-polytope, and constructing such realizations when they exist. This paper presents a characterization of realizable configurations for Guibas' conjecture based on work from the area of polytope convexity testing. Our approach to the decision and construction problems is a reduction to a linearinequality feasibility problem. The approach is also related to methods used for deciding if an arbitrary triangulation of a point set is a regular triangulation. We show two reductions, one based directly on a global convexity condition resulting in number of inequalities that is quadratic in the number of vertices of P, and one based on an equivalent local convexity condition resulting in a linear number of inequalities.