Planar Polyhedra

Abstract

The analysis presented in the previous chapters relies on an efficient implementation of the underlying polyhedral operations in order to attain acceptable performance. However, common implementations of convex polyhedra [14, 27, 93, 119] suffer from inherent scalability problems that mainly relate to the calculation of the join operation, which corresponds to the convex hull in the context of polyhedra. The classic approach for calculating the convex hull of two polyhedra is to convert the half-space representation using inequalities into the generator representation consisting of vertices, rays, and lines. Vertices are points in the polyhedron that cannot be represented by a convex combination of other points. Rays and lines are vectors that represent unidirectional and bidirectional trajectories, respectively, towards which the polyhedron extends to infinity. The convex hull of two input polyhedra can be calculated by converting both polyhedra into their generator representations, joining their sets of vertices, rays, and lines, and converting these three sets back into the half-space representation.