IV.4 - EXACT COMPUTATION OF 2-D INTERSECTIONS

Abstract

This chapter presents a technique for exact representation of the intersections between line segments in 2-D with vertices of a fixed-input resolution. The chapter explains how to represent a clipped line segment using bounded rational arithmetic, provided that both the clipping and clipped segments are defined by endpoints measured to fixed precision. The method for the exact computation of 2-D intersections allows a line segment to be clipped an arbitrary number of times without increasing storage requirements yet yielding the correct result. The output from a series of polygon-intersection operations must be composed of pieces of the line segments that make up the input polygons. A line subsegment is a portion of a line segment that is represented by: (1) two endpoints with integer coordinates whose precision is 15 bits and (2) two parameter values. The parameter value for the intersection point between two line subsegments is calculated by computing the intersection of the a priori line segments and then checking whether the intersection actually occurs within the subsegments. Given the parameter definition for an intersection point, cascaded clipping operations are supported by the ability to calculate the relative position of two such intersection points along a line segment.