Planar Projection of Polytopes using Hough Transforms

Abstract

Visualization of the state-space computed by statespace exploration tools for continuous and hybrid systems, is necessary for a better comprehension of the system dynamics. Tools like SpaceEx and XSpeed compute the state-space as a collection of polytopes. The visualization of the statespace, therefore requires an efficient algorithm for (1) Planar projection of high-dimensional polytopes and (2) Enumerating the vertices of the projection for plotting. In this paper, we propose a novel algorithm to compute a planar projection of a H-represented polytope. The proposed algorithm computes the projected polytope in V-representation and therefore can be readily visualized with plotting utilities. It is based on the support function representation of a polytope and Hough transforms. We believe that this work gives a new perspective of looking at the computation of planar projections, using classical results on Hough transforms in the domain of Computer Graphics. The limitation of the algorithm is that it incurs an approximation error, although the error can be measured and bounded. We show empirically that the performance of our algorithm is significantly better in comparison to known efficient algorithms of projection and enumeration, and therefore is viable when a limited approximation error is permissible.