Showing papers on "Regular polygon published in 1991"
TL;DR: A method for comparing polygons that is a metric, invariant under translation, rotation, and change of scale, reasonably easy to compute, and intuitive is presented.
Abstract: A method for comparing polygons that is a metric, invariant under translation, rotation, and change of scale, reasonably easy to compute, and intuitive is presented. The method is based on the L/sub 2/ distance between the turning functions of the two polygons. It 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 showing that the method produces answers that are intuitively reasonable are presented. >
733 citations
01 Jun 1991
TL;DR: A new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the listing of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension is presented.
Abstract: We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a)Virtually no additional storage is required beyond the input data.(b)The output list produced is free of duplicates.(c)The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.(d)The running time is output sensitive for nondegenerate inputs.(e)The algorithm is easy to parallelize efficiently.
For example, the algorithm finds thev vertices of a polyhedron inRd defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inRd, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inRd can be found inO(n2dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
581 citations
242 citations
TL;DR: The GEOMPACK package contains routines for constructing two- and three-dimensional Delaunay triangulations, decomposing a general polygonal region into simple or convex polygons, and other geometric algorithms, from which the mesh generation method is built and others can be implemented.
101 citations
100 citations
TL;DR: A two-dimensional automatic mesh generation technique has been proposed for composites, in which the second phase consisting of particulates or unidirectional fibers are randomly dispersed in the matrix to yield an aggregate of convex polygons.
89 citations
TL;DR: In this paper, Guibas et al. defined new operations on constructible functions (inverse or direct images, duality) and proved some theorems related to these operations (e.g., duality commutes to direct image).
81 citations
TL;DR: In this article, a negative shape is introduced in order to develop an algebraic system of geometric shapes within which one can add and subtract shapes exactly as one adds and subtracts within the integer number system.
Abstract: A new notion of negative shape is introduced in order to develop an algebraic system of geometric shapes within which one can add and subtract shapes exactly as one adds and subtracts within the integer number system. Concentrating on polygonal shapes in 2 dimensions, we show that this simple extension of our commonsense concept of geometric shapes opens up many new areas with a great potential for understanding and developing 2-dimensional geometry and geometric algorithms. In the course of this pursuit the concept of a new equivalence relation on convex polygons evolves that also appears to be significant in understanding convex polygons, particularly various symmetries in them. In constructing the algebraic system of shapes we use the Minkowski addition operation (in mathematical morphology dilation) as the composition Operation.
79 citations
01 Jun 1991
TL;DR: The star unfolding of a convex polytope with respect to a pointx on its surface is obtained by cutting the surface along the shortest paths fromx to every vertex, and flattening the surface on the plane.
Abstract: The star unfolding of a convex polytope with respect to a pointx on its surface is obtained by cutting the surface along the shortest paths fromx to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding:1.It does not self-overlap: it is a simple polygon.2.The ridge tree in the unfolding, which is the locus of points with more than one shortest path fromx, is precisely the Voronoi diagram of the images ofx, restricted to the unfolding.
These two properties permit conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well:?The construction of the ridge tree (in preparation for shortest-path queries, for instance) can be achieved by an especially simpleO(n2) algorithm. This is no worst-case complexity improvement, but a considerable simplification nonetheless.?The exact set of all shortest-path "edge sequences" on a polytope can be found by an algorithm considerably simpler than was known previously, with a time improvement of roughly a factor ofn over the old bound ofO(n7 logn).?The geodesic diameter of a polygon can be found inO(n9 logn) time, an improvement of the previous bestO(n10) algorithm.
Our results suggest conjectures on "unfoldings" of general convex surfaces.
79 citations
TL;DR: In this article, the authors investigated the number of lattice points in a convex body with a smooth boundary and nonzero Gaussian curvature, and established an asymptotic formula for the volume of the body.
Abstract: In this article we investigate the numberA(t) of lattice points in
$$\sqrt t$$
B whereB is a convex body in ℝ
s
(s≥3) which has a smooth boundary with nonzero Gaussian curvature throughout, andt is a large real parameter. We establish an asymptotic formulaA(t)=Vt
s/2+O(t
λ(s)) (V the volume ofB) which improves upon a classic result ofE. Hlawka [5].
57 citations
TL;DR: A data structure called a compact interval tree is introduced that supports common tangent computations, as well as the standard binary-search-based queries, in O(logn) time apiece and beats the lower bound for algorithms that store the polygons in independent arrays.
Abstract: In this paper, we investigate the problem of finding the common tangents of two convex polygons that intersect in two (unknown) points. First, we give a Θ(log2n) bound for algorithms that store the polygons in independent arrays. Second, we show how to beat the lower bound if the vertices of the convex polygons are drawn from a fixed set of n points. We introduce a data structure called a compact interval tree that supports common tangent computations, as well as the standard binary-search-based queries, in O(logn) time apiece. Third, we apply compact interval trees to solve the subpath hull query problem: given a simple path, preprocess it so that we can find the convex hull of a query subpath quickly. With O(nlogn) preprocessing, we can assemble a compact interval tree that represents the convex hull of a query subpath in O(logn) time. In order to represent arrangements of Lines implicitly, Edelsbrunner et al. used a less efficient structure, called bridge trees, to solve the subpath hull query problem. Our compact interval trees improve their results by a factor of O(logn). Thus, the present paper replaces the paper on bridge trees referred to by Edelsbrunner et al.
01 Jan 1991
TL;DR: In this article, the authors discuss formulas for calculating area of planar polygons and volume of polyhedra and highlight that these formulas are valid even for nonconvex polygons.
Abstract: Publisher Summary This chapter discusses formulas for calculating area of planar polygons and volume of polyhedra. It considers a planar polygon with vertices P 0 ,…, P n . There is a simple closed formula for the area of the polygon. If P n +1 = P 0 and if the points P 0 ,……, P n lie in the xy plane, then the formula for area of polygon can be derived from Green's theorem. If the points lie on some arbitrary plane perpendicular to a unit vector N , then the formula for area of the polygon is derived from Stokes theorem. These two formulas are valid even for nonconvex polygons. The chapter discusses formula for volume of a Polyhedron. It consider a polyhedron with planar polygonal faces S 0 ,…, S n . The chapter highlights that there is a simple closed formula for the volume of the polyhedron which is derived from Gauss Theorem.
TL;DR: In this paper, it was shown that within the point group of the polyhedron the symmetries spanned by the sets of vertices, faces and edges are also related.
Abstract: POLYHEDRAL cages and clusters are widespread in chemistry. Examples of fully triangulated polyhedra (deltahedra) are the skeletons of closo-boranes BnH2−n, many heteroboranes and transition-metal carbonyls1. Three-connected cages occur for carbon2,3 and in zeolites1. The numbers v, f and e of vertices, faces and edges of a convex polyhedron are related by Euler's theorem4,5 v+f=e + 2. Here we show that within the point group of the polyhedron the symmetries spanned by the sets of vertices, faces and edges are also related. We prove a general theorem relating these symmetries for convex polyhedra, and give further relations specific to deltahedra and 3-connected polyhedra. The latter extensions of Euler's theorem to point-group characters allow us to generate complete sets of internal vibrational coordinates from bond stretches for deltahedra, and to classify, from symmetry properties alone, the bonding or antibonding nature of molecular orbitals of 3-connected cages.
TL;DR: Shiffman as discussed by the authors proved that the intersection of a minimal annulus with any plane P, between P1 and P2, is a convex Jordan curve, and then used this convexity theorem to prove that every symmetry of the boundary of A extended to a symmetry of A.
Abstract: In 1956 M. Shiffman [17] proved several beautiful theorems concerning the geometry of a minimal annulus A whose boundary consists of two closed convex curves in parallel planes P1, P2. The first theorem stated that the intersection of A with any plane P , between P1 and P2, is a convex Jordan curve. In particular it follows that A is embedded. He then used this convexity theorem to prove that every symmetry of the boundary of A extended to a symmetry of A. In the case that ∂A consists of two circles Shiffman proved that A was foliated by circles in parallel planes. Earlier B. Riemann [15] described, in terms of elliptic functions, all minimal annuli in 3 that can be expressed as the union of circles in parallel planes (also see [3] for a
14 Aug 1991
TL;DR: It is said that a polygon P is immobilized by a set of points I on its boundary if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P.
Abstract: We say that a polygon P is immobilized by a set of points I on its boundary if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. Three immobilization points are always sufficient for a polygon with vertices in general positions, but four points are necessary for some polygons with parallel edges. An O(n log n) algorithm that finds a set of 3 points that immobilize a given polygon with vertices in general positions is suggested. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize. When the polytope has vertices in general position d+1 points are sufficient to immobilize.
TL;DR: Theoretical and experimental analyses show that the constants involved are small enough to make the algorithm competitive even for windows with four edges, and demonstrates the practicality of various ideas from computational geometry.
Abstract: We present an algorithm for clipping a polygon or a line against a convex polygonal window. The algorithm demonstrates the practicality of various ideas from computational geometry. It spendsO(logp) time on each edge of the clipped polygon, wherep is the number of window edges, while the Sutherland-Hodgman algorithm spendsO(p) time per edge. Theoretical and experimental analyses show that the constants involved are small enough to make the algorithm competitive even for windows with four edges. The algorithm enables image-space clipping against windows whose boundaries are convex spline curves. The paper contains detailed pseudo-code implementation of the algorithm and an adaptation of the simulation of simplicity method for handling degenerate cases.
01 Mar 1991
TL;DR: In this article, it was shown that if F has a transitive solvable group of isometries, then it is Ramsey and regular polygons in R 3 are Ramsey.
Abstract: A finite subset of a Euclidean space is called Ramsey if for each κ and each κ-coloring of a sufficiently dimensional Euclidean space E there is a monochromatic isometrical embedding from F to E. We show that if F has a transitive solvable group of isometries then it is Ramsey. In particular, regular polygons are Ramsey. We also show that regular polyhedra in R 3 are Ramsey
TL;DR: It is proved that for every n ⩾ 4 there is a convex n-gon such that the vertices of 2n − 7 vertex pairs are one unit of distance apart.
TL;DR: An explicit expression for the three-variable generating function P(x, y, z) = Sigma m>or=1, Sigma n>or = 1, Sigma r >or= 1 x2ny2mzrcn,m,r, where r is the number of convex polygons with horizontal width n, vertical height m and area r as discussed by the authors.
Abstract: An explicit expression is derived for the three-variable generating function P(x, y, z)= Sigma m>or=1, Sigma n>or=1 Sigma r>or=1 x2ny2mzrcn,m,r, where cn,m,r is the number of convex polygons with horizontal width n, vertical height m and area r.
TL;DR: An optimal EREW-PRAM parallel algorithm for the case when the input polygons are convex, and an essentially quadratic sequential algorithm in the case of arbitrary polygons (possibly with holes) are presented.
TL;DR: In this paper, it was shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves, without making use of advanced group-theoretical techniques.
Abstract: Motivated by a recently advanced conjecture on the ergodic properties of Quantum Systems, the problem of solving the Schrodinger equation for a free particle in a plane polygonal enclosure is revisited. It will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves, without making use of advanced group‐theoretical techniques. It turns out, inter alia, that these polygons, considered as classical billiards, are all and only those which are completely integrable in the sense of Arnold’s theorem.
TL;DR: A class of multi-valued mappings are exhibited which seem to impose themselves as natural generalizations of affine mappings, and which correspond to linearizing the support function of F which thus offers to be the most reasonable thing to do.
Abstract: One of the uses of the derivative of a nonlinear function F is to define first-order approximations of F by special functions, namely affine functions. Our aim is to extend this concept of first order approximations to the case when F is multi-valued; how should the concept of affinity be generalized for such functions? We first show why a mere copy of the single-valued case is hardly appropriate; a more sophisticated definition is needed for affinity. Restricting our study to convex compact valued F we consider various possibilities. We conclude by exhibiting a class of multi-valued mappings, which we call eclipsing, and which seem to impose themselves as natural generalizations of affine mappings. They correspond to linearizing the support function of Fwhich thus offers to be the most reasonable thing to do.
14 Aug 1991
TL;DR: The universal cover is a unifying framework that reveals connections between minimum paths under these three distance functions, as well as yielding simpler linear-time algorithms for shortest path trees and minimum link paths in simple polygons.
Abstract: In this abstract, we use the universal covering space of a surface to generalize previous results on computing paths in a simple polygon. We look at optimizing paths among obstacles in the plane under the Euclidean and link metrics and polygonal convex distance functions. The universal cover is a unifying framework that reveals connections between minimum paths under these three distance functions, as well as yielding simpler linear-time algorithms for shortest path trees and minimum link paths in simple polygons.
TL;DR: A dual approach to detect intersections of hyperplanes and convex polyhedra in arbitrary dimensions and these results are the first of their kind ford > 3.5 dimensions.
Abstract: This paper presents a dual approach to detect intersections of hyperplanes and convex polyhedra in arbitrary dimensions. In d dimensions, the time complexities of the dual algorithms are 0(2 d log n) for the hyperplane-polyhedron intersection problem, and O((2d) d- 1 logd- 1 n) for the polyhedron- polyhedron intersection problem. These results are the first of their kind for d > 3. In two dimensions, these time bounds are achieved with linear space and preprocessing. In three dimensions, the hyperplane-polyhedron intersection problem is also solved with linear space and preprocessing; quadratic space and preprocessing, however, is required for the polyhedron-polyhedron intersection problem. For general d, the dual algorithms require O(n 2d) space and preprocessing. All of these results readily extend to unbounded polyhedra.
TL;DR: A linear time sequential algorithm for finding a straight line that bisects two given disjoint convex polygons and can be generalized to other measures (for instance, perimeter) and other proportions of cutting.
TL;DR: A path-following algorithm for finding a solution to the nonlinear stationary point problem on an unbounded, convex, and pointed polyhedron and a condition under which the path of zeros converges to a solution is stated.
Abstract: A path-following algorithm is proposed for finding a solution to the nonlinear stationary point problem on an unbounded, convex, and pointed polyhedron. The algorithm can start at an arbitrary point of the polyhedron. The path to be followed by the algorithm is described as the path of zeros of some piecewise continuously differentiable function defined on an appropriate subdivided manifold. This manifold is induced by a generalized primal-dual pair of subdivided manifolds. The path of zeros can be approximately followed by dividing the polyhedron into simplices and replacing the original function by its piecewise linear approximation with respect to this subdivision. The piecewise linear path of this function can be generated by alternating replacement steps and linear programming pivot steps. A condition under which the path of zeros converges to a solution is also stated, and a description of how the algorithm operates when the problem is linear or when the polyhedron is the Cartesian product of a poly...
TL;DR: In this paper, it was shown that if K is a convex body in E, n 2, and p 0 is a point of K with the property that all bisections of K through p 0 are homothetic, then K is an Euclidean ball.
Abstract: We prove that if K is a convex body in E, n ^ 2, and p0 is a point of K with the property that all bisections of K through p0 are homothetic, then K is a Euclidean ball.
TL;DR: In this paper, the construction of rational basis functions for patchwork C0 approximation over a regular algebraic partition of a planar region is generalized to C1 approximation for convex quadrilateral elements.
Abstract: There is a well-defined construction of rational basis functions for patchwork C0 approximation over a regular algebraic partition of a planar region [2]. In this note the construction is generalized to C1 approximation for convex quadrilateral elements.
TL;DR: This work has revised and generalized one such algorithm to allow it to detect in C log n operations the intersection of two convex planar regions with piecewise curved boundaries, where n represents the number of curved segments.
TL;DR: In this paper, it was shown that the mean intercept polygon of a planar N-net system of straight lines is a convex 2N-sided polygon, and that the polygon can be constructed by an inversion process.
Abstract: With a view towards the characterization of microstructural anisotropy of fibrous materials, we have shown that the mean intercept figure for a planarN-net system of straight lines is a convex 2N-sided polygon. A very simple method of constructing the mean intercept figure for a planarN-net system is presented. It is shown, by example, that there is an inversion process by which one may construct a planarN-net system from its mean intercept polygon. The significance of these results with respect to the characterization of microstructural anisotropy of fibrous materials is discussed.