Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions

TL;DR

This paper addresses the problem of accurately rounding the intersection of lattice polygonal regions onto the integer lattice, ensuring inclusion properties and analyzing geometric characteristics to improve robustness in computational geometry.

Contribution

It introduces methods for inner and outer rounding of polygonal intersections on the lattice, with theoretical guarantees and analysis of geometric properties.

Findings

Proves inclusion properties for rounded intersections.

Analyzes Hausdorff distance and convexity of rounded regions.

Provides bounds on size and geometric features of the rounded polygons.

Abstract

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the --exact computation paradigm--[Yap et Dube] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions with integer vertices which respectively is included and contains the true intersection. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions.