Showing papers by "Herbert Edelsbrunner published in 1988"

Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms

Abstract: This paper describes a general purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task to provide a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those obtained without using it. We believe that this technique will become a standard tool in writing geometric software.

An optimal algorithm for intersecting line segments in the plane

Abstract: The authors present the first optimal algorithm for the following problem: given n line segments in the plane, compute all k pairwise intersections in O(n log n+k) time. Within the same asymptotic cost the algorithm will also compute the adjacencies of the planar subdivision induced by the segments, which is a useful data structure for contour-filling on raster devices. >

Probing convex polygons with X-rays

Abstract: An X-ray probe through a polygon measures the length of intersection between a line and the polygon. This paper considers the properties of various classes of X-ray probes, and shows how they interact to give finite strategies for completely describing convex n-gons. It is shown that $({{3n} / 2}) + 6$ probes are sufficient to verify a specified n-gon, while for determining convex polygons ${{(3n - 1)} / 2}$ X-ray probes are necessary and $5n + O(1)$ sufficient, with $3n + O(1)$ sufficient given that a lower bound on the size of the smallest edge of P is known.

Geometric probing

Abstract: We consider problems in geometric probing, the algorithmic study of determining a geometric structure or some aspect of that structure from the results of a mathematical or physical measuring device. A variety of problems from robotics, medical instrumentation, mathematical optimization, integral and computational geometry, graph theory, and other areas fit into this paradigm. Finger probes return the first point of intersection between a directed line l and an object P. Chapter 2 presents results on finger probing convex polygons. We consider related problems in higher dimensions and with different classes of objects. Hyperplane probes return the first hyperplane moving perpendicular to itself which is tangent to P. Chapter 3 discusses the duality relationship between finger and hyperplane probes. We establish the connection between hyperplane probes and certain algorithmic problems and consider the related silhouette and supporting line probe models. X-ray probes return the length of intersection between P and l. Chapter 4 surveys the field of tomography and presents results for x-ray probes, which was inspired by it. We give linear bounds on determination and verification with x-ray probes in two and higher dimensions. Half-space probes return the volume of intersection between a half-space h and P. Chapter 5 presents our linear determination and verification results for two dimensions and discussed the difficulties of determination in higher dimensions. Chapter 6 considers the power of infinite collections of these probes. We discuss Hammer's x-ray problem, presenting new proofs for convex polygons. Also, we discuss the combinatorial geometry problem of k-projections, which arises from aggregate probing. Finally, we consider other aggregate problems such as probing in rounds. Chapter 7 extends probing to an object which is not usually considered geometric. Cut-set probes return the size of a cut-set of a graph. We present surprising results using these to reconstruct and thus represent graphs. Each chapter concludes with relevant open problems.

The complexity of many faces in arrangements of lines of segments

Abstract: We show that the total number of edges of m faces of an arrangement of n lines in the plane is O(m2/3-dn2/3+2d + n), for any d > 0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of these m faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and, with high probability, its time complexity is within a log2n factor of the above bound. If instead of lines we have an arrangement of n line segments, then the maximum number of edges of m faces is Ogr;(m2/3-dn2/3+2d + na(n)logm), for any d > 0, where a(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and, with high probability, takes time that is within a log2n factor of the combinatorial bound.

Tetrahedrizing Point Sets in Three Dimensions

Abstract: This paper offers combinatorial results on extremum problems concerning the number of tetrahedra in a tetrahedrization of n points in general position in three dimensions, i.e. such that no four points are coplanar. It also presents an algorithm that in O(nlog n) time constructs a tetrahedrization of a set of n points consisting of at most 3n–11 tetrahedra.

Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms

Abstract: Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.

Searching for empty convex polygons

Abstract: A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r > 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.