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. 1

In this paper we present an efficient algorithm for the computation of the segment Voronoi diagram in two dimensions. Our algorithm can handle not only disjoint segments or segments that share endpoints, but also segments that may intersect at their interior. It is incremental and the expected cost of inserting n (possibly intersecting) sites (points or segments) is O((n + m) log 2 n), where m is the number of points of intersection of the (open) segments in the input site set. Finally, we describe the implementation of our algorithm, that uses techniques such as geometric filtering, and present experiments that show the robustness, efficiency and scalability of our implementation.

Let E r and E b be two sets of x-monotone and non-intersecting curve segments, E = E r [ E b and jEj = n. We give a new sweep-line algorithm that reports the k intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is O(n log n + k log log n) and it requires O(n) space.

In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals x for a quantity x and y for another quantity y, then, for every arithmetic operation fi, the set of possible values of x fi y also forms an interval; the operations leading from x and y to this new interval are called interval arithmetic operations. For addition and subtraction, corresponding interval operations consist of two corresponding operations with real numbers, so there is no hope of making them faster. The best known algorithms for interval multiplication consists of 3 real-number multiplications and several comparisons. We describe a new algorithm for which the average time is equivalent to using only 2 multiplications of real numbers. What is interval arithmetic. Many computer algorithms for processing real numbers have been designed to process measurement results. Measurements are never 100% precise; therefore, when after mea...

This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library Cgal.

Dedicated to the friends and families who blessed and supported me iv

In this paper we present a dynamic algorithm for the construction of the additively weighted Voronoi diagram of a set of weighted points in the plane. The novelty in our approach is that we use the dual of the additively weighted Voronoi diagram to represent it. This permits us to perform both insertions and deletions of sites easily. Given a set of n sites, among which h sites have a non-empty cell, our algorithm constructs the additively weighted Voronoi diagram of in O(nT (h) + h log h) expected time, where T (k) is the time to locate the nearest neighbor of a query site within a set of k sites with a non-empty cell. Deletions can be performed for all sites whether or not their cell is empty. The space requirements for the presented algorithm is O(n). Our algorithm is simple to implement and experimental results suggest an O(n log h) behavior.

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2 Quadrics 4 2.1 Projective Space Preliminaries................. 4 2.2 Quadrics and Associated Matrices............... 4