Efficient algorithms for line and curve segment intersection using restricted predicates

We consider whether restricted sets of geometric predicates support efficie nt algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, propos ed by Preparata and others as a way to guide the search for efficient algor ithms that can be implemented in more realistic computational models than t he 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 comp arisons and above/below tests for endpoints, and intersection tests, must t ake at least Omega (n root k) time, There are optimal O(n log n + k) algori thms that use a higher-degree test comparing x coordinates of an endpoint a nd intersection point; for line segments we show that this test can be simu lated 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 pseudo-segment intersection, in which the segments are colored red and blue so that there are no red/red or blue/blue crossings. (C) 2000 Elsevier Science B.V. All r ights reserved.