Showing papers on "Computational geometry published in 1985"

Computational geometry. an introduction

Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."

Voronoi diagram in the laguerre geometry and its applications

Abstract: We extend the concept of Voronoi diagram in the ordinary Euclidean geometry for n points to the one in the Laguerre geometry for n circles in the plane, where the distance between a circle and a po...

Some dynamic computational geometry problems

Abstract: We consider some problems in computational geometry when every one of the input points is moving in a prescribed manner.

Movable Separability of Sets

Abstract: Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane or polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, presents new results, and provides a list of open problems and suggestions for further research.

Optimal algorithms for symmetry detection in two and three dimensions

Abstract: Exact algorithms for detecting all rotational and involutional symmetries in point sets, polygons and polyhedra are described. The time complexities of the algorithms are shown to be θ (n) for polygons and θ (n logn) for two- and three-dimensional point sets. θ (n logn) time is also required for general polyhedra, but for polyhedra with connected, planar surface graphs θ (n) time can be achieved. All algorithms are optimal in time complexity, within constants.

Geometric Algorithms for Digitized Pictures on a Mesh-Connected Computer

Abstract: Although mesh-connected computers are used almost exclusively for low-level local image processing, they are also suitable for higher level image processing tasks. We illustrate this by presenting new optimal (in the O-notational sense) algorithms for computing several geometric properties of figures. For example, given a black/white picture stored one pixel per processing element in an n × n mesh-connected computer, we give ?(n) time algorithms for determining the extreme points of the convex hull of each component, for deciding if the convex hull of each component contains pixels that are not members of the component, for deciding if two sets of processors are linearly separable, for deciding if each component is convex, for determining the distance to the nearest neighboring component of each component, for determining internal distances in each component, for counting and marking minimal internal paths in each component, for computing the external diameter of each component, for solving the largest empty circle problem, for determining internal diameters of components without holes, and for solving the all-points farthest point problem. Previous mesh-connected computer algorithms for these problems were either nonexistent or had worst case times of ?(n2). Since any serial computer has a best case time of ?(n2) when processing an n × n image, our algorithms show that the mesh-connected computer provides significantly better solutions to these problems.

Intersection and Closest-Pair Problems for a Set of Planar Discs

Abstract: Efficient algorithms for detecting intersections and computing closest neighbors in a set of circular discs, are presented and analyzed. They adapt known techniques for solving these problems for sets of points or of line segments. The main portion of the paper contains the construction of a generalized Voronoi diagram for a set of n (possibly intersecting) circular discs in time $O(n\log ^2 n)$, and its applications.

Practical use of Bucketing Techniques in Computational Geometry

Abstract: Techniques for using “buckets” to improve the efficiency of several computational-geometrical algorithms are described, together with examples illustrating the practical importance of the bucketing techniques. Specifically, they are applied to the problems of minimum-weight perfect matchings in the plane, two-dimensional Voronoi diagrams, point location and range search in the plane, and shortest paths in networks.

Computational Geometry and Motion Planning

Abstract: Research and development work in robotics and industrial automation has created a need for good motion planning algorithms for collision avoidance; this in turn has stimulated research on the inherent computational geometry of motion planning. The purpose of this paper is to survey some recent algorithms and complexity results for motion planning, placing particular emphasis on computational geometry issues. A brief discussion of a variety of the many different types of problems that arise in the context of motion planning is included, as many of these problems involve geometric questions. The main focus of the paper, however, is on the following problem and its variants: Given descriptions of an object and a collection of obstacles in 2 or 3-dimensional space, and given the initial and desired final configuration for the object, find a collision-free trajectory that moves the object to its goal or determine that no such trajectory exists.

An Isothetic View of Computational Geometry

Abstract: We survey a number of results, some new, for isothetic polygons in the plane. Specifically we consider intersection, convexity, combinational, clustering and visibility problems. These problems occur in image processing, VLSI design and layout, circuit testing, transaction systems, and pattern recognition.

A fast algorithm for the Boolean masking problem

Abstract: An algorithm is presented for the calculation of Boolean combinations between layers of a VLSI circuit layout. Each layer is assumed to contain only polygons, which are specified by their edges; the output is also polygonal. The algorithm runs in O((n + k)(r + log n)) time and O(nr) space, where n is the total number of edges on all layers, k is the number of edge intersections, and r is the number of layers. Also a number of restrictions on the general problem are discussed which lead to substantial improvements in the time bounds. It is proved that when the polygons are presented using a hierarchical description language the problem becomes NP-hard. Finally how this approach can be used to solve the i-contour problem of computational geometry and the hidden-line-elimination problem of computer graphics is discussed.

The Beta2-spline: A Special Case of the Beta-spline Curve and Surface Representation

Abstract: Simpler equations and computationally more efficient algorithms make the Beta2-spline technique easier to understand and useful to the designer.

Geometric optimization of manipulator structures for working volume and dexterity

Abstract: Broadly speaking, the regional structure of a manipulator, which consists of the inboard three joints and the members associated with them, determines the workspace shape and volume. The orientation structure, which, for a six degree of freedom manipulator, consists of the three furthest outboard joints and members, determines the geometric dexterity or orientation potential of the manipulator. It is possible, using straightforward geometric arguments, to determine the optimal dimensions of the regional structure for a given total length. By the use of the spherical counterpart of Grashof's theorem formulated by Freudenstein, it is possible also to show that there is an optimum geometry of the orientation structure.

Distance problems in computational geometry with fixed orientations

Abstract: In computational geometry, problems involving only rectilinear objects with edges parallel to the x -and y-axes have attracted great attention. They are often easier to solve than the same problems for arbitrary objects, and solutions are of high practical value, for instance in VLSI design. This is because in VLSI design technology requirements often dictate the use of only two orthogonal orientations for the boundary edges of objects as well as wires.The restriction on the boundary edges motivates the study of rectilinear objects, while the restriction on wires brings the focus on the well-known L1-metric (the Manhattan distance). In short, given the two orthogonal orientations, both the shape of objects and the distance function are determined in a natural way.More recent VLSI fabrication technology is capable of creating edges and wires in both the orthogonal and diagonal orientations. This motivates us to study more general polygons, and to generalize the distance concept to the case where any fixed set of orientations is allowed. We introduce a family of naturally induced metrics, and the subsequent generalization of geometrical concepts. A shortest connection between two points is in this case a path composed of line segments with only the given orientations. We derive optimal solutions for various basic planar distance problems in this setting, such as the computation of a Voronoi diagram, a minimum spanning tree, and the (minimum and maximum) distance between two convex polygons. Many other theoretically interesting and practically relevant problems remain to be solved. In particular, the new family of metrics may help bridge the gap between the L1- and the L2-metrics, as those are the limiting cases for two and infinitely many regularly distributed orientations.

Translating Polygons in the Plane

Abstract: Let P = (p1,...,pn) and Q = (q1,...,qm) be two simple polygons with non-intersecting interiors in the plane specified by their cartesian coordinates in order. Given a direction d we can ask whether P can be translated an arbitrary distance in direction d without colliding with Q. It has been shown that this problem can be solved in time proportional to the number of vertices in P and Q. Here we present a new and efficient algorithm for determining all directions in which such movement is possible. In designing this algorithm a partitioning technique is developed which might find applications when solving other geometric problems. The algorithm utilizes several tools and concepts (e.g. convex hulls, point-location, weakly edge-visible polygons) from the area of computational geometry.

Some applications of resultants to problems in computational geometry

Abstract: Resultants were originally developed in the 18th and 19th centuries to solve certain simple algebraic problems. Here we shall present some applications of resultants to several important problems in computational geometry, including the implicitization, inversion, and intersection of parametric rational polynomial curves and surfaces.

New algorithms for special cases of the hidden line elimination problem

Abstract: Hidden line elimination is a well-known problem in computer graphics and many practical solutions have been proposed. Only recently the problem has been studied from a theoretical point of view, taking asymptotic worst-case time- and spacebounds into account. Here we study three special cases of increasing difficulty and generality of the hidden line elimination problem. Applying some methods from computational geometry these problems can be solved with better worst-case bounds than those of the best known algorithms for the general problem.

A worst-case efficient algorithm for hidden-line elimination †

Abstract: Many practical algorithms for hidden-line and surface elimination in a 2-dimensional projection of a 3-dimensional scene have been proposed. However surprisingly little theoretical analysis of the algorithms has been carried out. Indeed no non-trivial lower bounds for the problem are known. We present a plane-sweep-based hidden-line-elimination algorithm for 2-dimensional projections of scenes consiting of arbitrary polyhedra. It requires, in the worst case0(n log n) space and 0((n + k) log2 n) time, where n is the number of edges in the 3-dimensional scene, and k is the number of edge intersections in the specific projection.

A computational geometry approach to clustering problems

Abstract: This paper deals with the relationship between cluster analysis and computational geometry describing clustering strategies using a Voronoi diagram approach in general and a line separation approach to improve the efficiency in a special case. We state the following theorems : The set of all centralized 2-clusterings (S1,S2) of a planar point set S with |S1|=a and |S2|=b is exactly the set of all pairs of labels of opposite Voronoi polygons va(S1,S) and vb(S2,S) of Va(S) and Vb(S) respectively.An optimal centralized 2-clustering [centralized divisive hierarchical 2- clustering] can be constructed in O(n n1/2 log2n + UF(n) n n1/2 + PF(n)) [O(n n1/2 log3n + UF(n) n n1/2 + PF(n)) respectively] steps with PF(n) and UF(n) being the time complexity to compute and update a given clustering measure f.

Geometric Data Structures for Computer Graphics

Abstract: The area of Computational Geometry deals with the study of algorithms for problems concerning geometric objects like e.g. lines, polygons, circles etc. in the plane and in higher dimensional space. Since its introduction in 1976 by Shamos[17] the field has developed rapidly and nowadays there are even special conferences and journals devoted to the topic. A list of publications by Edelsbrunner and van Leeuwen[8] collected in 1982 already contained over 650 papers. And this number has rapidly increased since then.

Optimal Parallel Algorithms for Selection, Sorting and Computing Convex Hulls

Abstract: Parallel algorithms are described for three problems that are often to be solved in computational geometry, namely selection, sorting and computing convex hulls. For a problem of size n, all three algorithms use n 1-e processors where 0

Expected Time Analysis of Algorithms in Computational Geometry

Abstract: We give a brief inexhaustive survey of recent results that can be helpful in the expected time analysis of algorithms in computational geometry. Most fast average time algorithms use one of three principles: bucketing, divide-and-conquer (merging), or quick elimination (throw-away). To illustrate the different points, the convex hull problem is taken as our prototype problem. We also discuss searching, sorting, finding the Voronoi diagram and the minimal spanning tree, identifying the set of maximal vectors, and determining the diameter of a set and the minimum covering sphere.

Parallel Computational Geometry (Extended Abstract)

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Clustering Geometric Objects and Applications to Layout Problems

Abstract: This paper deals with the relationship between cluster analysis and computational geometry describing clustering strategies using a Voronoi diagram approach in general and a line separation approach to improve the efficiency in a special case. We state the following theorems: 1 The set of all centralized 2-clusterings (S1,S2) of a planar point set S with |S1| = a and |S2| = b is exactly the set of all pairs of labels of opposite Voronoi polygons va (S1,S) and vb(S2,S) of Va (S) and Vb(S) respectively. 2 An optimal centralized 2-clustering [centralized divisive hierarchical 2-clustering]3,can be constructed in 0(n√n log2n + UF(n) • n√n + PF (n))[0(N√B=N • log n + UF(n) • n√n + PF(n)) respectively] steps with PF(n) and UF(n) being the time complexity to compute and update a given clustering measure f. Applications to layout problems (design of assembly lines; VLSI-placement problems and board design) as well as image understanding problems (clustering of sets of geometric objects as points, edges, polygons etc.) will be given in the talk.

A unifying approach for a class of problems in the computational geometry of polygons

Abstract: A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is O(n2), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and rerporting) in O(n logn), O(n) or O(logn) time, depending on the nature of a specialized “selection” function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of “locality” in set mappings contribute significantly to the derivation of the results.