Double Precision Geometry: A General Technique for Calculating Line and Segment Intersections Using Rounded Arithmetic

We show for the rst time how to reduce the cost of performing speciic geometric constructions by using rounded arithmetic instead of exact arithmetic. Exploiting a property of oating point arithmetic called monotonicity, a new technique, double precision geometry, can replace exact arithmetic with rounded arithmetic in any eecient algorithm for computing the set of intersections of a set of lines or line segments. This technique reduces the complexity of any such line or segment arrangement algorithm by a constant factor. In addition, double precision geometry reduces by a factor of N the complexity of rendering segment arrangements on a 2 N by 2 N integer grid such that output segments have grid points as endpoints. 1 Introduction The theory of computational geometry provides us with a large number of useful geometric algorithms. However, it is not easy to create eecient and reliable programs from these algorithms because they are either based on innnite precision arithmetic or the precision required is not explicitly stated. The standard practice among developers of geometric programs is to use nite precision oating point arithmetic as if it were innnite precision real arithmetic. Unfortunately, geometric programs are particularly sensitive to numerical errors because of they way they closely couple symbolic and numerical data. The common experience is that all oating point geometric programs fail on a small subset of their inputs and that these failures are diicult to analyze or avoid. There is clearly a practical need for a theory of robust geometric algorithms, algorithms which can be safely implemented with rounded nite precision arithmetic. This paper describes part of such a theory and its expected payoo, robust approximate geometric algorithms with faster running times than the best exact versions. A theory of robust geometry requires an understanding of the precision requirements of exact algorithms: the num