Showing papers on "Regular polygon published in 1990"
01 Nov 1990
TL;DR: An algorithm for compositing a combination of density clouds and contour surfaces used to represent a scalar function on a 3-D volume subdivided into convex polyhedra, which provides a method for visualizing such data sets.
Abstract: We present an algorithm for compositing a combination of density clouds and contour surfaces used to represent a scalar function on a 3-D volume subdivided into convex polyhedra. The scalar function is interpolated between values defined at the vertices, and the polyhedra are sorted in depth before compositing. For n tetrahedra comprising a Delaunay triangulation, this sorting can always be done in O(n) time. Since a Delaunay triangulation can be efficiently computed for scattered data points, this provides a method for visualizing such data sets. The integrals for opacity and visible intensity along a ray through a convex polyhedron are computed analytically, and this computation is coherent across the polyhedron's projected area.
215 citations
Book•
01 Jul 1990
TL;DR: In this article, a collision-free path algorithm for convex polygonal bodies is presented, which is based on characterizing the volume swept by a body as it is translated and rotated as a generalized cone, and determining under what conditions one generalized cone is a subset of another.
Abstract: Free space is represented as a union of (possibly overlapping) generalized cones. An algorithm is presented which efficiently finds good collision-free paths for convex polygonal bodies through space littered with obstacle polygons. The paths are good in the sense that the distance of closest approach to an obstacle over the path is usually far from minimal over the class of topologically equivalent collision-free paths. The algorithm is based on characterizing the volume swept by a body as it is translated and rotated as a generalized cone, and determining under what conditions one generalized cone is a subset of another.
200 citations
01 Feb 1990
TL;DR: A methodology for computing the distance between objects in three-dimensional space is presented, where the convex polytope is replaced by a general convex set, avoiding the errors caused by the usualpolytope approximations and actually reducing the overall computational time.
Abstract: A methodology for computing the distance between objects in three-dimensional space is presented. The convex polytope is replaced by a general convex set, avoiding the errors caused by the usual polytope approximations and actually reducing the overall computational time. The basic algorithm is a simple extension of the polytope distance algorithm described by E.G. Gilbert et al. (1988). It utilizes the support mappings of the sets representing the objects. A calculus for evaluating these mappings that allows the extended algorithm to be applied to a rich family of nonpolytopal objects is presented. While the convergence of the algorithm is not finite, it is fast and an effective stopping condition that guarantees the accuracy of the numerical solution is available. Extensive numerical experiments support the claimed efficiency. >
195 citations
TL;DR: How the algorithm can be modified to give an O( n \alpha ( n ) ) algorithm for a class of dynamic programming problems satisfying convex quadrangle inequalities results in faster algorithms for a number of problems arising in molecular biology, speech recognition, and geology.
Abstract: An $O( m\alpha ( n ) + n )$ time algorithm is given for finding row-maxima and minima in totally monotone partial $n \times n$ matrices. As a result, faster algorithms are obtained for some optimization problems concerning distance and visibility between vertices of two convex polygons. Also shown is how the algorithm can be modified to give an $O( n \alpha ( n ) )$ algorithm for a class of dynamic programming problems satisfying convex quadrangle inequalities. This results in faster algorithms for a number of problems arising in molecular biology, speech recognition, and geology.
111 citations
01 Jan 1990
TL;DR: A method for comparing polygons that has these properties and works for both convex and nonconvex polygons and runs in time O(mn log mn) where m is the number of vertices in one polygon and n is the size of the polygons in the other.
Abstract: : Model-based recognition is concerned with comparing a shape A, which is stored as a model for some particular object, with a shape B, which is found to exist in an image. If A and B are close to being the same shape, then a vision system should report a match and return a measure of how good that match is. To be useful this measure should satisfy a number of properties, including: (1) it should be a metric, (2) it should be invariant under translation, rotation, and change-of-scale, (3) it should be reasonably easy to compute, and (4) it should match our intuition (i.e., answers should be similar to those that a person might give). We develop a method for comparing polygons that has these properties. The method works for both convex and nonconvex polygons and runs in time O(mn log mn) where m is the number of vertices in one polygon and n is the number of vertices in the other. Some examples are presented that show the method produces answers that are intuitively reasonable.
97 citations
TL;DR: An explicit expression for the perimeter and area generating function G(y, z)= Sigma n>or=2 Sigma m>or =1 cn,mynZm, where cn is the number of row-convex polygons with area m and perimeter n.
Abstract: An explicit expression is obtained for the perimeter and area generating function G(y, z)= Sigma n>or=2 Sigma m>or=1 cn,mynZm, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.
70 citations
01 Jul 1990
TL;DR: It is shown that the optimal δS-approximation of an n-gon P by an axes-parallel rectangle can be found in time O(log3n) by a nested binary search algorithm.
Abstract: We consider the approximation of convex polygons by simpler figures such as rectangles, circles, or polygons with fewer edges. As distance measures for figures A, B we use either the area of the symmetric difference δS(A, B) or the Hausdorff-distance δH(A, B). It is shown that the optimal δS-approximation of an n-gon P by an axes-parallel rectangle can be found in time O(log3n) by a nested binary search algorithm. With respect to δ H pseudo-optimal algorithms are given, i.e. algorithms producing a solution whose distance to P differs from the optimum only by a constant factor. We obtain algorithms of runtimes O(n) for approximation by rectangles and O(n3log2n) for approximation by k-gons (k
48 citations
TL;DR: It is proved that given n⩾3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n−9 sides, and with not less than 3n−6 distinct slopes.
44 citations
TL;DR: In this paper, the authors show that a number of geometric problems can be solved on a √n × √ n mesh-connected computer in O(√n) time, which is optimal to within a constant factor.
Abstract: We show that a number of geometric problems can be solved on a √n × √n mesh-connected computer (MCC) inO(√n) time, which is optimal to within a constant factor, since a nontrivial data movement on an MCC requires Ω(√n) time. The problems studied here include multipoint location, planar point location, trapezoidal decomposition, intersection detection, intersection of two convex polygons, Voronoi diagram, the largest empty circle, the smallest enclosing circle, etc. TheO(√n) algorithms for all of the above problems are based on the classical divide-and-conquer problem-solving strategy.
42 citations
TL;DR: In this article, it was shown that if every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and 0 ϵ R(A), then the strong limt → ∞xt exists and belongs to A−10.
39 citations
TL;DR: In this article, the authors investigated the property of largeness for monotonic core allocation in a TU cooperative game with populationN and showed that if the upper bound depends only upon the size of the coalition and varies monotonically as the size increases, then there exists a monotone core allocation meeting this system of upper bounds, and applied this result to the provision of a public good problem.
Abstract: In a TU cooperative game with populationN, a monotonic core allocation allocates each surplusv (S) among the agents of coalitionS in such a way that agenti's share never decreases when the coalition to which he belongs expands We investigate the property of largeness (Sharkey [1982]) for monotonic cores We show the following result Given a convex TU game and an upper bound on each agent' share in each coalition containing him, if the upper bound depends only upon the size of the coalition and varies monotonically as the size increases, then there exists a monotonic core allocation meeting this system of upper bounds We apply this result to the provision of a public good problem
TL;DR: In this paper, it was shown that the sum of the winding and tangent numbers is independent of the point with respect to which they are taken, and equals the rotation number.
Abstract: The winding and rotation numbers for closed plane polygons and curves appear in various contexts. Here alternative definitions are presented, and relations between these characteristics and several other integer-valued functions are investigated. In particular, a point-dependent "tangent number" is defined, and it is shown that the sum of the winding and tangent numbers is independent of the point with respect to which they are taken, and equals the rotation number.
TL;DR: Kleene’s algorithm applied to this closed semiring solves the problem of determining whether a directed graph with two-dimensional labels has a zero-sum cycle or not, and is shown to run in polynomial time in the special cases of graphs with one-dimensional names, BTTSP, and graphs with bounded labels.
Abstract: Two natural operations on the set of convex polygons are shown to form a closed semiring; the two operations are vector summation and convex hull of the union. Various properties of these operations are investigated. Kleene’s algorithm applied to this closed semiring solves the problem of determining whether a directed graph with two-dimensional labels has a zero-sum cycle or not. This algorithm is shown to run in polynomial time in the special cases of graphs with one-dimensional labels, BTTSP (Backedged Two-Terminal Series-Parallel) graphs, and graphs with bounded labels. The undirected zero-sum cycle problem and the zero-sum simple cycle problem are also investigated.
TL;DR: In this article, it was shown that the diameter of a reduced body in Euclidean space is not greater than √ 2Δ(R), and that the perimeter is at most (2+½π)Δ (R).
Abstract: A convex bodyR of Euclideand-spaceEd is called reduced if there is no convex body properly contained inR of thickness equal to the thickness Δ(R) ofR. The paper presents basic properties of reduced bodies inE2. Particularly, it is shown that the diameter of a reduced bodyR⊂E2 is not greater than √2Δ(R), and that the perimeter is at most (2+½π)Δ(R). Both the estimates are the best possible.
01 May 1990
TL;DR: The maximin placement of a convex polygon P inside a polygon Q is shown, and the dynamic Voronoi diagram of rigidly moving sets of n points is investigated, showing the combinatorial complexity of this canonical dynamic diagram.
Abstract: This paper considers the maximin placement of a convex polygon P inside a polygon Q, and introduce several new static and dynamic Voronoi diagrams to solve the problem. It is shown that P can be placed inside Q, using translation and rotation, so that the minimum Euclidean distance between any point on P and any point on Q is maximized in O(m4n λ16(mn) log mn) time, where m and n are the numbers of edges of P and Q, respectively, and λ16(N) is the maximum length of Davenport-Schinzel sequences on N alphabets of order 16. If only translation is allowed, the problem can be solved in O(mn log mn) time. The problem of placing multiple translates of P inside Q in a maximum manner is also considered, and in connection with this problem the dynamic Voronoi diagram of k rigidly moving sets of n points is investigated. The combinatorial complexity of this canonical dynamic diagram for kn points is shown to be O(n2) and O(n3k4 log* k) for k = 2, 3 and k ≥ 4, respectively. Several related problems are also treated in a unified way.
TL;DR: It is shown that, if the vertex-edge visible pairs are found, the maximum matching and the maximum independent set of the bipartite graph derived from the chords of a simple rectilinear polygon can be found in linear time without constructing the bipartsite graph.
Abstract: An O(n log log n) algorithm is proposed for minimally rectangular partitioning a simple rectilinear polygon. For any simple rectilinear polygon P, a vertex-edge visible pair is a vertex and an edge that can be connected by a horizontal or vertical line segment that lies entirely inside P. It is shown that, if the vertex-edge visible pairs are found, the maximum matching and the maximum independent set of the bipartite graph derived from the chords of a simple rectilinear polygon can be found in linear time without constructing the bipartite graph. Using this algorithm, the minimum partition problem for convex rectilinear polygons and vertically (horizontally) convex rectilinear polygons can be solved in O(n) time. >
TL;DR: In this paper, a solution of the problem is given based on a decomposition of the set of all schedules into local regions in which the optimization problem is convex and the criterion functions are related to network flow problems.
TL;DR: A novel technique for proving lower bounds in parallel computation based on mapping any algorithm for the problem being considered to an algorithm for another problem, for which a good lower bound is known.
Journal Article•
01 May 1990
TL;DR: This study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion planning problem, by describing algorithms which will run faster when certain implicit implicit parameters of the input are bounded away from 1.
Abstract: For compact Euclidean bodies P, Q, we define λ(P, Q) to be smallest ratio r/s where r > 0, s > 0 satisfy sQ′ ⊆ P ⊆ rQ″. Here sQ denotes a scaling of Q by factor s, and Q′, Q″ are some translates of Q. This function λ gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies are homothetic if one can be obtained from the other by scaling and translation).For integer k ≥ 3, define λ(k) to be the minimum value such that for each convex polygon P there exists a convex k-gon Q with λ(P, Q) ≤ λ(k). Among other results, we prove that 2.118… ≤ λ(3) ≤ 2.25 and λ(k) = 1 + T(k-2). We give an O(n2 log2 n) time algorithm which for any input convex n-gon P, finds a triangle T that minimizes λ(T, P) among triangles. But in linear time, we can find a triangle t with λ(t, P) ≤ 2.25.Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion planning problem. In each case, we describe algorithms which will run faster when certain implicit slackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.
TL;DR: Three approximation algorithms to cover a rectilinear polygon that is neither horizontally nor vertically convex by rectangles are developed, and experimental results indicate that the algorithms with complexity O(n/sup 2/) and O( n/sup 4/) often obtain optimal or near-optimal covers.
Abstract: Three approximation algorithms to cover a rectilinear polygon that is neither horizontally nor vertically convex by rectangles are developed. All three guarantee covers that have at most twice as many rectangles as in an optimal cover. One takes O(n log n) time, where n is the number of vertices in the rectilinear polygon. The other two take O(n/sup 2/) and O(n/sup 4/) time. Experimental results indicate that the algorithms with complexity O(n/sup 2/) and O(n/sup 4/) often obtain optimal or near-optimal covers. >
TL;DR: It is shown that no strategy can achieve complete reconstruction using less than 3 V − 3 measurements, and duality implies that the same reconstruction performance is achieved when probing with a composite finger probe.
TL;DR: Algorithms to reconstruct the planar cross-section of a simply connected object from data points measured by rays, showing that the information provided by the rays is crucial when solving 2-dimensional reconstruction problems.
16 Jun 1990
TL;DR: The concepts of an augmented EGI, cross-adjacency, and fictitious EGI points are introduced and utilized in the recovery of the adjacency of faces in the reconstruction of the polyhedron.
Abstract: The extended Gaussian image (EGI) of a polyhedron is a compact representation of a polyhedral object and is unique for convex polyhedra. A set of equations is developed, the solution of which reconstructs the polyhedron. The concepts of an augmented EGI, cross-adjacency, and fictitious EGI points are introduced and utilized in the recovery of the adjacency of faces. An example is included to illustrate the use of this technique. >
TL;DR: The perimeter generating function derived recently by Brak et al. (1990) for row-convex polygons on the square lattice is generalized to the rectangular lattice as mentioned in this paper.
Abstract: The perimeter generating function derived recently by Brak et al (1990) for row-convex polygons on the square lattice is generalized to the rectangular lattice.
TL;DR: In this paper, the theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas, where points are computed as the common zeros of a set of polynomials.
Abstract: The theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas in a systematic way. The points are computed as the common zeros of a set of polynomials. These polynomials constitute a canonical basis for the real ideal belonging to the formula. For a symmetric formula they can be chosen to span a representation of the symmetry group. For regions with high symmetry and formulas of low degree this leads to only a few distinct possibilities and the existence of the corresponding formulas is easily checked. In general, however, the canonical basis is not completely determined by symmetry and degree, and a suitable choice has to be made. The application to regular polygons in two dimensions is discussed and formulas are presented for polygons with three, five, six, seven, and eight vertices, with varying degrees of exactness (up to order 27 for the hexagon).
TL;DR: The questions of the existence of such 3-polytopes with exactly two types of edges are solved and the cardinalities of all classes are determined.
TL;DR: The singularity expansion method parameters (natural frequencies, natural modes, and coupling coefficients) for regular planar polygonal loops (equilateral triangle, square, regular pentagon, and regular hexagon) are computed and compared to those determined for the regular 60-gon (pseudo-circular) loop as discussed by the authors.
Abstract: The singularity expansion method parameters (natural frequencies, natural modes, and coupling coefficients) for regular planar polygonal loops (equilateral triangle, square, regular pentagon, and regular hexagon) are computed and compared to those determined for the regular 60-gon (pseudo-circular) loop Common characteristics of these geometries are compared and contrasted with respect to their use as radiating antenna structures or scatterers The approach results in new insights for these canonical loop structures >
Patent•
28 Aug 1990
TL;DR: In this article, an autocollimator for projecting collimated light beams onto a facet of a polygon, a matrix array detector for detecting and transducing into angle measuring data the location of the focused light reflected from the polygon facets, a processor and angle error memory unit, for computing the difference between the angles of any given facet and an adjacent facet relative to a respective angle of an ideal regular polygon and a control unit for processing and compensating for the angle data to yield a corrected angle measurement.
Abstract: An optical angle measuring apparatus for measuring the angular displacement of a polygon with light reflecting facets, for use, e.g., in machine tools. The apparatus comprises an autocollimator for projecting collimated light beams onto a facet of a polygon, a matrix array detector for detecting and transducing into angle measuring data the location of the focused light reflected from the polygon facets, a processor and angle error memory unit; for computing the difference between the angles of any given facet and an adjacent facet relative to a respective angle of an ideal regular polygon, and a control unit for processing and compensating for the said angle data to yield a corrected angle measurement. The corrected angle measurement is displayed on display means. A method for measuring the angular displacement of an object relative to a reference position employing the apparatus is also disclosed.