In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use expression compiler called EXPCOMP. EXPCOMP supports all common operations +; \Gamma; \Delta; =; p . Applying a new semi-static filter, EXPCOMP combines the speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow. The resulting programs show a very good runtime behaviour. 1 Introduction When computer scientists design geometric algorithms, they usually assume the availability of exact arithmetic on real numbers. Since no computer directly provides exact arithmetic on real numbers, programmers implementing these algorithms must find some substitution. Quite commonly, they resort to floating-point arithmetic due to its support by hardand software as well as its co...

We are given a transportation line where displacements happen at a bigger speed than in the rest of the plane. A shortest time path is a path between two points which takes less or equal time than any other. We consider the time to follow a shortest time path to be the time distance between the two points. In this paper, we give a simple algorithm for computing the Time Voronoi Diagram, that is, the Voronoi Diagram of a set of points using the time distance.

In this paper we talk about a new ecient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various oating-point lters together with arbitrary precision packages, we develop an easy-to-use expression compiler called EXPCOMP. EXPCOMP supports all common operations +; ; ; =; p . Applying a new semi-static lter, EXPCOMP combines the speed of static lters with the power of dynamic lters. The lter stages deal with all kinds of oating-point exceptions, including underow. The resulting programs show a very good runtime behaviour. Keywords: oating-point lter, exact arithmetic, expression compiler 1. Introduction When computer scientists design geometric algorithms, they usually assume the availability of exact arithmetic on real numbers. Since no computer directly provides exact arithmetic on real numbers, programmers implementing these algorithms must nd some substitution. Quite commonly, they resort to oating-point arithmetic due t...

We present a general framework for computing two-dimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3-space, which implements a divide-and-conquer algorithm. A straightforward application of the divide-andconquer approach for Voronoi diagrams yields highly inefficient algorithms. We show that through randomization, the expected running time is near-optimal (in a worst-case sense). We believe this result, which also holds for general envelopes, to be of independent interest. We describe the interface between the construction of the diagrams and the underlying construction of the envelopes, together with methods we have applied to speed up the (exact) computation. We then present results, where a variety of diagrams are constructed with our implementation, including power diagrams, Apollonius diagrams, diagrams of line segments, Voronoi diagrams on a sphere, and more. In all cases the implementation is exact and can handle degenerate input.

Algorithmic problems on proximity and location under metric constraints

Robert A. Katz ###################################### ################################ (Under the direction of Professor Stephen M. Pizer) This work presents a method for quantifying important form cues in 2D polygonal models. These cues are derived from multi-scale medial analysis, and they also draw upon several new form analysis methods developed in this work. Among these new methods are a means of using the Blum Medial Axis Transform for stable computing, a method to decompose objects into a set of parts that includes the ambiguity we find in human perception of objects, and the simultaneous application of both internal and external medial representations of objects. These techniques are combined to create a local saliency measure, a global saliency measure and a measure of object complexity. This work also demonstrates a new approach to simplifying complex polygonal models in order to accelerate interactive rendering. I propose that simplifying a model's form, based on how it is perceived and processed by the human visual system, offers the potential for more effective simplifications. In this research, I suggest a means of understanding object simplification in these perceptual terms by creating a perceptually based scheduling of simplification operations as well as perceptual measures of the degree of simplification and the visual similarity between a simplified object and its original. A new simplification scheme is based on these measures, and then this perceptual scheme is compared via examples to a geometric simplification method. iv To my grandmother She fueled my curiosity and inspired me by graduating college at the age of 75, and she taught me to always see the goodness of people. May she celebrate this work with me, wherever she is. ACKNOWLEDGEMENTS Than...