Static Analysis Yields Efficient Exact Integer Arithmetic for Computational Geometry

Abstract

Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extended-precision arithmetic is required. We describe static-analysis techniques that reduce the performance cost of exact integer arithmetic used to implement geometric algorithms. We have used the techniques for a number of examples, including line-segment intersection in two dimensions, Delaunay triangulations, and a three-dimensional boundary-based polyhedral modeller. In general, the techniques are appropriate for algorithms that use primitives of relatively low algebraic total degree, e.g., those involving flat objects (points, lines, planes) in two or three dimensions. The techniques have been packaged in a preprocessor for reasonably convenient use.