Showing papers on "Computational geometry published in 1984"
TL;DR: It is suggested that B6zier curves should be considered as a single type of curve, but the design of these curves should not be confused with other types of curves, which are known to have different properties.
839 citations
01 Dec 1984
TL;DR: Three techniques in computational geometry are explored: scaling solves a problem by viewing it at increasing levels of numerical precision; activation is a restricted type of update operation, useful in sweep algorithms; the Cartesian tree is a data structure for problems involving maximums and minimums.
Abstract: Three techniques in computational geometry are explored: Scaling solves a problem by viewing it at increasing levels of numerical precision; activation is a restricted type of update operation, useful in sweep algorithms; the Cartesian tree is a data structure for problems involving maximums and minimums. These techniques solve the minimum spanning tree problem in Rk1 and Rk
579 citations
TL;DR: The state of the art of computational geometry is surveyed, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms.
Abstract: We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas—convex hulls, intersections, searching, proximity, and combinatorial optimizations—are discussed. Seven algorithmic techniques—incremental construction, plane-sweep, locus, divide-and-conquer, geometric transformation, prune-and-search, and dynamization—are each illustrated with an example. A collection of problem transformations to establish lower bounds for geo-metric problems in the algebraic computation/decision model is also included.
271 citations
TL;DR: It is shown that CSG trees representing null objects may be reduced to null trees through the use of a new concept called primitive redundancy, and that, on average, tree reduction can be done efficiently by a new technique called spatial localization.
Abstract: Constructive solid geometry (CSG) is the primary scheme used for representing solid objects in many contemporary solid modeling systems. A CSG representation is a binary tree whose nonterminal nodes represent Boolean operations and whose terminal nodes represent primitive solids. This paper deals with algorithms that operate directly on CSG representations to solve two computationally difficult geometric problems—null-object detection (NOD) and same-object detection (SOD). The paper also shows that CSG trees representing null objects may be reduced to null trees through the use of a new concept called primitive redundancy, and that, on average, tree reduction can be done efficiently by a new technique called spatial localization. Primitive redundancy and spatial localization enable a single complex instance of NOD to be converted into a number of simpler subproblems and lead to more efficient algorithms than those previously known.
111 citations
TL;DR: In this paper, vector techniques and elimination methods are combined to help resolve some classical problems in computer aided geometric design, such as implicit polynomial equation of the curve (implicitization), and finding the parameter value corresponding to the coordinates of a point known to lie on the curve.
96 citations
01 Dec 1984
TL;DR: Efficient computation of the 'cut' guaranteed by the classical 'Ham Sandwich theorem', faster preprocessing time for polygon retrieval, and theoretical improvements to a problem of intersecting lines and points posed by Hopcroft.
Abstract: For any set X of points (in any dimension) and any k = 1,2, ..., we introduce the concept of the k-hull of X. This unifies the well-known notion of 'convex hulls' with the notion of 'centers' recently introduced by F.F. Yao. The concept is intimately related to some other concepts (k-belts, k-sets) studied by Edelsbrunner, Welzl, Lovasz, Erdos and others. Several computational problems related to k-hulls are studied here. Some of our algorithms are of interest in themselves because of the techniques employed; in particular, the 'parametric' searching technique of Megiddo is used in a nontrivial way. We will also extend Megiddo's technique to Las Vegas algorithms. Our results have applications to a variety of problems in computational geometry: efficient computation of the 'cut' guaranteed by the classical 'Ham Sandwich theorem', faster preprocessing time for polygon retrieval, and theoretical improvements to a problem of intersecting lines and points posed by Hopcroft.
83 citations
TL;DR: Implementations yielding maximal throughput are given for solving dynamic versions of convex hull, inclusion, range and intersection search, planar point location, intersection, triangulation, and closest-point problems.
Abstract: This paper describes systolic algorithms for a number of geometric problems. For the sake of realism we restrict our investigation to one-dimensional arrays whose communication links with the outside are located at the end cells. Implementations yielding maximal throughput are given for solving dynamic versions of convex hull, inclusion, range and intersection search, planar point location, intersection, triangulation, and closest-point problems.
75 citations
Journal Article•
38 citations
01 Oct 1984-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: A study of some algorithmic problems involved in windowing a picture is offered and some methods from computational geometry are exploited to store the picture in a computer such that those line segments inside or partially inside of a window can be determined efficiently.
Abstract: Windowing a two-dimensional picture means to determine those line segments of the picture that are visible through an axis-parallel window. A study of some algorithmic problems involved in windowing a picture is offered. Some methods from computational geometry are exploited to store the picture in a computer such that (1) those line segments inside or partially inside of a window can be determined efficiently, and (2) the set of those line segments can be maintained efficiently while the window is moved parallel to a coordinate axis and/or it is enlarged or reduced.
TL;DR: A thorough treatment of the Cayley method of elimination is given, with particular emphasis on its implementation to solve problems in computational geometry.
TL;DR: This work derives concepts for such space-sweep algorithms that yield an efficient solution to the problem of solving any set operation of two convex polyhedra that matches the best known time bound of O(n log n).
Abstract: Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding general space-sweep algorithms for geometric problems in 3- space are known. We derive concepts for such space-sweep algorithms that yield an efficient solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Our solution matches the best known time bound of O(n log n), where n is the combined number of vertices of the two polyhedra.
01 Dec 1984
TL;DR: In this article, it was shown that it is possible to compute all k intersections between n arbitrary line segments in time O(k + f(n)), where f is a subquadratic function of n.
Abstract: This paper settles a long-standing open question of computational geometry: Is it possible to compute all k intersections between n arbitrary line segments in time linear in k? We answer this question affirmatively by presenting the first algorithm with a running time of the form O(k + f(n)), where f is a subquadratic function of n. The function f we achieve is actually quasi-linear in n, which makes our algorithm the most efficient to date for each value of k. To obtain this result we must turn away from traditional, sweep-line-based schemes. Instead, we introduce a new hierarchical strategy for dealing with segments without ever reducing the dimensionality of the problem. This framework is used to solve other related problems. In particular, we are able to present the first subquadratic algorithm for counting intersections (as opposed to reporting each of them explicitly), and we give the first optimal algorithm for computing the intersections of a line arrangement with a query segment. Using duality arguments we also present an improved algorithm for a point enclosure problem.
01 Jan 1984
TL;DR: The GMSolid system for modeling solids is being enhanced by the addition of free-form surfaces represented as B-splines as discussed by the authors, which provides a convenient mathematical foundation for many of the computational geometry algorithms needed in a solid modeler.
Abstract: The GMSolid system for modeling solids is being enhanced by the addition of free-form surfaces represented as B-splines. The major reason for this enhancement is that a wide variety of automotive parts cannot be modeled satisfactorily using existing GMSolid methods based on quadric surfaces. Free-form surfaces, which can be conformed to twisting, nonquadric surface regions, are required for efficient and accurate modeling. Representing free-form surfaces as B-splines provides a convenient mathematical foundation for many of the computational geometry algorithms needed in a solid modeler.
01 Aug 1984-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: In this paper, a combinatorial investigation of sets of moving points, that is, dynamical sets of points of points, was initiated, where the points are assumed to be instantiated at the same instant of time, each with some constant speed.
Abstract: This paper initiates the combinatorial investigation of sets of moving points, that is dynamical sets of points. In this, the first step, the points are assumed to be instantiated at the same instant of time, each with some constant speed. It is observed how dynamical point sets give rise to half-lines or rays in a one-dimensional higher space-time model. This in turn leads to efficient solutions for many problems concerning one-dimensional dynamical sets, occurring naturally in our framework, using computational geometry techniques. However, this initial investigation raises more questions than it answers; for example, is there an algorithm which determines for a two-dimensional dynamical set all possible point coincidences in time better than O(n2)?
TL;DR: The results show that divide-and-conquer can be used in place of line sweep, without additional asymptotic cost, for some geometrical problems.
Abstract: In this correspondence we reconsider three geometrical problems for which we develop divide-and-conquer algorithms. The first problem is to find all pairwise intersections among a set of horizontal and vertical line segments. The second is to report all points enclosures occurring in a mixed set of points and rectangles, and the third is to find all pairwise intersections in a set of isooriented rectangles. We derive divide-and-conquer algorithms for the first two problems which are then combined to solve the third. In each case a space-and time-optimal algorithm is obtained, that is O(n) space and O(n log n + k) time, where n is the number of given objects and k is the number of reported pairs. These results show that divide-and-conquer can be used in place of line sweep, without additional asymptotic cost, for some geometrical problems. This raises the natural question: For which class of problems are the line sweep and divide-and-conquer paradigms interchangeable?
TL;DR: The two-dimensional convex hull algorithms of Graham, Jarvis, Eddy, and Akl and Toussaint are tested on four different planar point distributions and the Graham algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull.
Abstract: The two-dimensional convex hull algorithms of Graham, Jarvis, Eddy, and Akl and Toussaint are tested on four different planar point distributions. Some modifications are discussed for both the Graham and Jarvis algorithms. Timings taken of FORTRAN implementations indicate that the Eddy and Akl-Toussaint algorithms are superior on uniform distributions of points in the plane. The Graham algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull.
TL;DR: The purpose of this note is to give an efficient algorithm for checking whether two planar figuresP andQ are similar when the figures are no longer constrained to be polygons, and to prove that the algorithm is optimal.
Abstract: Two planar figures aresimilar if a scaled version of one of them can be moved so that it coincides with the second figure. The problem of checking whether two planar figures are similar is relevant to both computational geometry and pattern recognition. An efficient algorithm is known for checking whether two polygonsP andQ are similar(1) The purpose of this note is to give an efficient algorithm for checking whether two planar figuresP andQ are similar when the figures are no longer constrained to be polygons. We give anO(n logn) time algorithm for solving this problem when each figure consists of a collection of (possibly intersecting) straight line segments, circles, and ellipses. Our algorithm can easily be modified for figures which include other geometric patterns as well. We also prove that our algorithm is optimal.
Proceedings Article•
01 Sep 1984TL;DR: This technique, widely applicable to computational geometry, is useful whenever objects are represented by quadtrees and only a portion of the data structure is required.
Abstract: This technique, widely applicable to computational geometry, is useful whenever objects are represented by quadtrees and only a portion of the data structure is required.
DOI•
01 Jan 1984
TL;DR: This work derives concepts for three-dimensional space-sweep algorithms that yield an elegant solution to the problem of solving any set operation of two convex polyhedra that matches the best known time bound of O(n log n).
Abstract: Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding three-dimensional space-sweep algorithms for geometric problems in 3-space are known, however. We derive concepts for such space-sweep algorithms that yield an elegant solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Moreover, our solution matches the best known time bound of O(n log n) where n is the combined number of corners of the two polyhedra.
11 Apr 1984
TL;DR: Several key-problems of the classical part of computational geometry which exhibit strong interrelations are presented, and a unified view of the problems is stressed, and the general ideas behind the methods that solve them are worked out.
Abstract: Computational geometry, considered a subfield of computer science, is concerned with the computational aspects of geometric problems. The increasing activity in this rather young field made it split into several reasonably independent subareas. This paper presents several key-problems of the classical part of computational geometry which exhibit strong interrelations. A unified view of the problems is stressed, and the general ideas behind the methods that solve them are worked out.
01 Mar 1984
TL;DR: A comparison of the two approaches is made and the geometric approach is shown to be the more beneficial although somewhat less systematic.
Abstract: This paper presents two approaches to the solution of the coordinate transformation problem for industrial manipulators: Geometric and Mathematical. A comparison of the two approaches is made and the geometric approach is shown to be the more beneficial although somewhat less systematic.
TL;DR: It is shown that the average case performance of the divide‐and‐conquer approach to solve the planar closest pair problem is proportional to the number of poins being examined.
Abstract: Bentley proposed a divide‐and‐conquer approach to solve the planar closest pair problem. In this paper, we shall show that the average case performance of this algorithm is proportional to the number of poins being examined.
TL;DR: The idea for a Special Issue of Transactions on Graphics devoted to Computational Geometry arose from conversations between myself, Leo Guibas, and Jorge Nievergelt at Xerox Palo Alto Research Center in the spring of 1983, and it was therefore decided to publish two full Special Issues and include two papers in the next regular issue of TOG.
Abstract: The idea for a Special Issue of Transactions on Graphics devoted to Computational Geometry arose from conversations between myself, Leo Guibas, and Jorge Nievergelt at Xerox Palo Alto Research Center in the spring of 1983. We felt that it was time to attempt to bridge the historical divide between the two schools of computational geometry that had emerged during the 1970's, and that TOG was an appropriate forum. The call for papers produced more contributions than we had expected, many of high quality, and some lengthy. It was therefore decided to publish two full Special Issues (April 1984 and this one) and include two papers in the next regular issue of TOG {January 1985). We would like to thank contributors and referees for their help and valued advice. The earliest reference to computational geometry that I have been able to discover appears to be Minsky and Papert 's 1969 book "Perceptrons," subtitled "An Introduction to Computational Geometry." [1] Unfortunately the term is not defined, nor is it mentioned in the index, or as far as I recall, elsewhere in the book, and we are left to draw our own conclusions as to what the authors intended the term to mean! Independently, while at Cambridge University, I had decided that using computers to perform geometric calculations required a fresh approach to geometry and I defined computational geometry in 1971 as