Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

Abstract

This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex notions are those of boundary and separating edges. When performing proofs about this algorithm, questions of symmetry appear and this exposition attempts to give an account of how these symmetries can be handled. All this work relies on formal developments made with Coq and the mathematical components library. An Abstract Description of TriangulationGiven a set of points, a triangulating algorithm returns a collection of triangles that must cover the space between these points (the convex hull), have no overlap, and such that all the points of the input set are vertices of at least one triangle. When the input points represent obstacles, the triangulation can help construct safe routes between these obstacles, thanks to Delaunay triangulations and Voronoï diagrams.

Safe Smooth Paths Between Straight Line Obstacles

“…The computation of sub-areas of the plane is already studied in work on convex hulls [8] and triangulations [4,1]. The algorithm we use to decompose the workspace into cells relies extensively on the orientation predicate taken from the work of Knuth [6].…”

Polygon Merge: A Geometric Algorithm Verified Using PVS

“…A method to enhance the automatability of proofs of geometric algorithms targets TLA+2 [6]. A different approach was used to apply Coq to the verification of a triangulation algorithm [1].…”