Efficient algorithms for line and curve segment intersection using restricted predicates

Abstract

We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM. Suppose that n (pseudo-)segments have k intersections at which they cross. We show that intersection algorithms for monotone curves that use only comparisons and above/below tests for end-points, and intersection tests, must take at least Ω(n√k) time. There are optimal O(n log n+k) algorithms that use a higher-degree test comparing x coordinates of an endpoint and intersection point; for line segments we show that this test can be simulated using CCW() tests with a logarithmic loss of efficiency. We also give an optimal O(n log n+k) algorithms for red/blue line and curve segment intersection, in which the segments are colored red and blue so that there are no red/red or blue/blue crossings.