Showing papers on "Regular polygon published in 1993"

The visibility complex

Abstract: We introduce the visibility complex of a collection O of n pairwise disjoint convex objects in the plane. This 2 dimensional cell complex may be considered as a generalization of the tangent visibility graph of 0. Its space complexity k is proportional to the size of the tangent visibility graph. We give an O(nlog n+k) algorithm for its construction. Furthermore we show how the visibility complex can be used to compute the view from a point or a convex object with respect to O in O(m log n) time, where m is the size of the view. The view from a point is a generalization of the visibility polygon of that point with respect to O.

On the form factor between two polygons

Abstract: Form factors are used in radiosity to describe the fraction of diffusely reflected light leaving one surface and arriving at another. They are a fundamental geometric property used for computation. Many special configurations admit closed form solutions. However, the important case of the form factor between two polygons in three space has had no known closed form solution. We give such a solution for the case of general (planar, convex or concave, possibly containing holes) polygons. CR Categories and Subject Descriptors: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism ‐ Radiosity ; J.2 [Physical Sciences and Engineering]: Engineering.

Minimal knotted polygons on the cubic lattice

Abstract: The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.

Constructing strongly convex approximate hulls with inaccurate primitives

Abstract: The first half of this paper introducesEpsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives. Epsilon Geometry is based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases. The second half of the paper introduces the notion of a (−ɛ)-convex polygon, a polygon that remains convex even if its vertices are all arbitrarily displaced by a distance ofɛ of less, and proves some interesting properties of such polygons. In particular, we prove that for every point set there exists a (−ɛ)-convex polygonH such that every point is at most 4ɛ away fromH. Using the tools of Epsilon Geometry, we develop robust algorithms for testing whether a polygon is (−ɛ)-convex, for testing whether a point is inside a (−ɛ)-convex polygon, and for computing a (−ɛ)-convex approximate hull for a set of points.

Separating and intersecting spherical polygons: computing machinability on three-, four-, and five-axis numerically controlled machines

Abstract: We consider the computation of an optimal workpiece orientation allowing the maximal number of surfaces to be machined in a single setup on a three-, four-, or five-axis numerically controlled machine. Assuming the use of a ball-end cutter, we establish the conditions under which a surface is machinable by the cutter aligned in a certain direction, without the cutter's being obstructed by portions of the same surface. The set of such directions is represented on the sphere as a convex region, called the visibility map of the surface. By using the Gaussian maps and the visibility maps of the surfaces on a component, we can formulate the optimal workpiece orientation problems as geometric problems on the sphere. These and related geometric problems include finding a densest hemisphere that contains the largest subset of a given set of spherical polygons, determining a great circle that separates a given set of spherical polygons, computing a great circle that bisects a given set of spherical polygons, and finding a great circle that intersects the largest or the smallest subset of a set of spherical polygons. We show how all possible ways of intersecting a set of n spherical polygons with v total number of vertices by a great circle can be computed in O(vn log n) time and represented as a spherical partition. By making use of this representation, we present efficient algorithms for solving the five geometric problems on the sphere.

CHAPTER 1.11 – Special Convex Bodies

Abstract: Publisher Summary This chapter discusses special convex bodies like simplices, ellipsoids, centrally symmetric bodies, and bodies of constant width. A simplex in Ed is defined as the convex hull of d + 1 affinely independent points. Leichtweiss used the infimum of the quotients of circumradius and inradius over classes of affinely equivalent convex bodies, Santalo and Eggleston used the area and a certain width integral of a plane convex body in order to characterize simplices. For studying ellipsoids from the view point of convexity one has a good starting point in the surveys of Bonnesen and Fenchel, Gruber and Hobinger, and Petty, which summarize results on characterizations of ellipsoid by means of sections and projections; by extremal properties; using the group of projectivities or subgroups thereof; and also on ellipsoids in Minkowski geometry and Hilbert geometry. Furthermore, there are a lot of geometric inequalities yielding ellipsoids as extremal bodies. Also, approximation of convex bodies by convex polytopes may lead to ellipsoids as extremal bodies. Ellipsoids may also be characterized by the product of mean width and surface area or by the quotient of inradius and circumradius. Characterizations of Euclidean spaces or, in infinite dimensions, of inner product spaces also give characterizations of ellipsoids. Also, Klee and Bolker have characterized centrally symmetric polytopes by intersection and projection properties.

Space structures with non-periodic subdivisions of polygonal faces

Abstract: A family of space structures having subdivided faces, where such faces are subdivided into rhombii in non-periodic arrangements. The rhombii are derived from regular planar stars with n vectors, and the source space structures are composed of regular polygons. The family includes: globally symmetric structures where the fundamental region is subdivided non-periodically, or globally asymmetric structures composed of regular polygons which are subdivided non-periodically or asymmetrically. The rhombii can be further subdivided periodically or non-periodically. The family further includes all regular polyhedra in the plane-faced and curve-faced states, regular tessellations, various curved polygons, cylinders and toroids, curved space labyrinths, and regular structures in higher-dimensional and hyperbolic space. The structures can be isolated structures or grouped to fill space. Applications include architectural space structures, fixed or retractable space frames, domes, vaults, saddle structures, plane or curved tiles, model-kits, toys, games, and artistic and sculptural works realized in 2- and 3-dimensions. The structures could be composed of individual units capable of being assembled or disassembled, or structures which are cast in one piece, or combination of both. Various tensile and compressive structural systems, and techniques of triangulation could be used as needed for stability.

The writhe of a self-avoiding polygon

Abstract: We discuss the writhe of a self-avoiding polygon on a lattice, as a geometrical measure of its entanglement complexity. We prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon, and use Monte Carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils.

Intrinsic volumes and lattice points of crosspolytopes

Abstract: Hadwiger showed by computing the intrinsic volumes of a regular simplex that a rectangular simplex is a counterexample to Wills' conjecture for the relation between the lattice point enumerator and the intrinsic volumes in dimensions not less than 441. Here we give formulae for the volumes of spherical polytopes related to the intrinsic volumes of the regular crosspolytope and of the rectangular simplex. This completes the determination of intrinsic volumes for regular polytopes. As a consequence we prove that Wills' conjecture is false even for centrally symmetric convex bodies in dimensions not less than 207.

Modeling of packing problems

Abstract: In this paper the modeling of packing problems (cutting and allocation problems) is investigated. Especially, linear mixed-integer models are proposed for two-dimensional packing problems of convex and non-convex polygons. The models are based on the so-called (outer) hodograph and the inner hodograph. Rotations of the polygons are not allowed. The principles of modeling in the two-dimensional case can be applied alsofor three- and higher dimensional packing problems

Compatible tetrahedralizations

Abstract: We give some special-case tetrahedralization algorithms. We first consider the problem of finding a tetrahedralization compatible with a fixed triangulation of the boundary of a polyhedron. We then adapt our solution to the related problem of compatibly tetrahedralizing the interior and exterior of a polyhedron. We also show how to tetrahedralize the region between nested convex polyhedra with O(n log n) tetrahedra and no Steiner points.

A compaction algorithm for non-convex polygons and its application

Abstract: Given a two dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied “forces.” Compaction can be modeled as a motion of the polygons that reduces the value of some linear functional on their positions. Optimal compaction, planning a motion that finds the global minimum reachable value, is shown to be NP-complete. We give a compaction algorithm that finds a local minimum by direct calculation of the new polygon positions via linear programming. We also consider the related problem of separating overlapping polygons using a minimal amount of motion and show it to be NP-complete. A locally optimum version of this problem is solved using a slight modification of the compaction algorithm. The compaction algorithm and the separation algorithm have been applied to marker making: the task of packing polygonal pieces on a sheet of cloth of fixed width so that total length is minimized. The compaction algorithm has improved cloth utilization of human generated pants markers. The separation algorithm together with a database of human-generated markers can be used to automatically generate markers that are close to human performance.

Hadwiger-Levi’s Covering Problem Revisited

Abstract: A well-known problem of discrete geometry is due to Hadwiger (1957), (1960) and Levi (1955). The following conjecture concerning this problem was published by Hadwiger (1957), (1960) and also by Gohberg and Markus (1960): Any convex body of E d , d ≥ 1 (i.e. any compact convex subset of the d-dimensional Euclidean space E d with non-empty interior) can be covered by 2 d smaller homothetic bodies and equality is attained only for d-dimensional parallelotopes. This conjecture has stimulated a lot of research in geometry. To survey the basic results we need some simple definitions.

On Functional Determinants of Laplacians in Polygons and Simplices

Abstract: The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} \lambda_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons.

Visibility graphs of staircase polygons with uniform step length

Abstract: Let Γn denote the collection of visibility graphs of staircase polygons (orthogonal convex fans) which consist of n−1 horizontal steps of arbitrary lengths. We show that for n≥7 there are graphs in...

Probing polygons minimally is hard

Abstract: Let Γ be a set of convex unimodal polygons in fixed position and orientation. We prove that the problem of determining whether k finger probes are sufficient to distinguish among the polygons in Γ is NP-complete for two types of finger probes. This implies that the same results hold for most interesting classes of polygons on which finger probes can be used.

CHAPTER 5.1 – Integral Geometry

Abstract: Publisher Summary This chapter discusses integral geometry. Integral geometry is concerned with the study, computation, and application of invariant measures on sets of geometric objects. It has its roots in some questions on geometric probabilities. Integral geometry is closely connected to the geometry of convex bodies. The chapter describes some notation, concepts, and results concerning the main spaces occurring in the integral geometry of Euclidean spaces. The most familiar type of integral-geometric formula is the intersection of a fixed and a moving geometric object. For example, the principal kinematic formula for convex bodies provides an explicit expression for the measure of all positions of a moving convex body, in which it meets a fixed convex body K. Crofton's intersection formula does the same for the invariant measure of the set of all k-dimensional flats meeting a convex body. The functionals of convex bodies appearing in the results, the quermassintegrals or intrinsic volumes, can replace the characteristic functions in the integrations with respect to invariant measures. The resulting formulae can be further generalized, as they are valid in local versions—namely, for curvature measures.

An Optimal Algorithm for Roundness Determination on Convex Polygons

Abstract: In tolerancing, the Out-Of-Roundness factor determines the relative circularity of planar shapes. The measurement of concern in this work is the Minimum Radial Separation, as recommended by the American National Standards Institute (ANSI). Here presented is a further clarification of the complexity of a previously presented algorithm of Van-Ban Le and D. T. Lee to determine the Minimum Radial Separation of simple polygons, which is found to be Θ(n2). Secondly, an optimal O(n) time algorithm to compute the Minimum Radial Separation of convex polygons is presented, which represents not only a factor n improvement over the previously best known algorithm, but also a factor of log n improvement over Le and Lee's conjectured complexity for the problem.

Computing Minkowski sums

Abstract: Minkowski sums are a useful tool for applications of geometric modelling and the sum of two given regions can be regarded as the region generated by sweeping a given region along the other. Though the sums have often been applied, and their theoretical properties studied, little has appeared concerning efficient calculation in the general case where both summands are non-convex. This thesis studies Minkowski sums of regular, bounded and path connected polygons, plane algebraic curves, and regular bounded and path-connected polyhedra in detail, discussing algorithms for their computation and establishing sharp output size estimates. It also studies some of the applications of Minkowski sums. In the polygonal case, we study the Minkowski sums of regular, bounded and path connected polygons in detail, discussing algorithms for their computation and establishing a sharp output size estimate. We also provide examples and discuss the observed time complexity and contrast it with the possible worst case complexity of the algorithms. We also present an extension of the main algorithm to the case where the polygons are unbounded. The Polyhedral algorithms presented are extensions of the planar case. In the case of algebraic curves, we show that the boundary of the Minkowski sum consists of portions of the "envelope" of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. We also formulate a simple numerical procedure for the case of polynomial parametric curves, based on constructing the Gauss maps of the given curves and using them to identify "corresponding" curve segments that are to be summed. We then present a method to extract the true boundary from the sum of these segments. We have used Minkowski sums to develop a new, simple, and efficient primitive for interpolating polyhedra: a Parameterized Interpolating Polyhedron, or PIP for short. PIPs are easily specified and edited by providing their initial and final shapes, which may be any polyhedra, and need not have corresponding boundary elements, nor be convex. We provide simple and efficient algorithms, for computing PIPs, which can be efficiently displayed using standard depth-buffer hardware.

Using representations of the dihedral groups in the design of early vision filters

Abstract: An approach to filter design which uses only the geometrical properties of the grid on which the filters are defined is proposed. This leads naturally to the study of the dihedral groups (i.e., the groups that leave the regular polygons invariant). It is shown how the representations of these groups can be used to develop algorithms that simplify a whole class of operators. The transforms developed are noncommutative generalizations of the discrete Fourier transform. The results are illustrated with some examples from the design of early vision filters. >

Device for processing three-dimensional image information, with extraction of significant lines.

Abstract: An image is defined, in correspondence with a predetermined polyhedral meshing of a portion of the space, by a numerical value for each vertex of the meshing. Polygon processing includes receiving a representation of a plane polygon with p vertices, p respective values for these p vertices, and a threshold. In correspondence with the polygon, polygon processing supplies a list of oriented segments linking points of the edges of the polygon which are interpolated as being equal to the threshold, with a predetermined direction convention. Polyhedron processing includes receiving a representation of a polyhedron with the values at its vertices and a threshold. Polyhedron processing then invokes polygon processing for all the faces of the polyhedron, which supplies a list of oriented segments. Polyhedron processing then associates with the polyhedron a list of closed and oriented cycles, constituted by segments from the said list of segments. Main processing includes sequentially presenting a plurality of polyhedrons of the polyhedrons meshing for the polyhedron processing, with the values associated with their vertices and a threshold. The cycles thus obtained belong to an iso-surface of the image for the threshold value.