Polygon Merge: A Geometric Algorithm Verified Using PVS

Abstract

Geometric algorithms can present significant challenges for formal methods. We describe the formalization and verification of an algorithm posing such challenges. Given two overlapping polygons A and B, the Polygon Merge algorithm derives a new polygon whose edges are outermost, partial edges of A and B. The algorithm has been verified to satisfy correctness criteria expressed as two point-set properties. We have constructed a rigorous proof using the PVS interactive theorem prover along with support from the NASA PVS Library. While the algorithm itself is fairly compact, its proof was a moderately complicated undertaking. Owing to the complexity of reasoning about the spatial relationships of two different polygons, the proof required hundreds of supporting lemmas. Most of the definitions and lemmas introduced were needed to build a body of deductive artifacts sufficiently robust for this problem domain. Many of these artifacts will be generally useful for working with polygons, line segments and 2D vectors. Accordingly, we are distilling these products into a reusable PVS library.