Shortest Path Geometric

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. This exponential growth arises when the algorithms are cascaded: the output of one algorithm becomes the input to the next. Cascading is a signiicant problem in practice. We propose a geometric rounding technique: shortest path rounding. Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to any algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact oating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on oating point input coordinates; the exact output coordinates can be rounded to accurate oating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware oating point operation.