Algebraic methods and arithmetic filtering for exact predicates on circle arcs
Olivier Devillers, Alexandra Fronville, Bernard Mourrain, Monique Teillaud · 2000
The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs.We focus on the comparison of the abscissae of two intersection points of circle arcs, which is known to be a difficult predicate involved in the computation of arrangements of circle arcs.We present an algorithm for deciding the x-order of intersections from the signs of the coefficients of a polynomial, obtained by a general approach based on resultants.This method allows the use of efficient arithmetic and filtering techniques leading to fast implementation as shown by the experimental results. I. INTRODUCTIONImplementing geometric algorithms is difficult because the decisions made by such algorithms are taken on the basis of simple geometric questions, called predicates, solved by the evaluation of continuous functions subject to rounding errors, though the algorithms are basically of combinatorial and discrete nature.For example, the sweep line paradigm is a combinatorial algorithm relying on predicates such as x-comparisons.The use of floating point arithmetic to evaluate predicates often produces inconsistencies.For instance, plane sweep algorithms, which are basic tools in computational geometry, are known to be very sensitive to numerical errors: when computing arrangements of curves, a plane sweep algorithm needs to sort intersection points between curves by x coordinates, and if, due to erroneous numerical computations, the x comparison test is not transitive, the algorithm may crash.To cope with this problem, people may either work on the *This research