Showing papers on "Regular polygon published in 1980"

Computational geometry and convexity

Abstract: The purpose of this dissertation is two-fold: To assert the power of convexity as a crucial factor of efficiency in computational geometry and to show how non-convex design can also benefit from this feature. Most of the recent results in computational geometry have relied on the attribute of convexity, and have failed to generalize to arbitrary designs. To remedy this flaw, one general approach consists of decomposing the objects into convex pieces, then applying the procedures to each part. We study the problem of finding minimal convex decompositions in two and three dimensions. Among our major results are an O(n + N('3)) dynamic-programming algorithm for producing minimal decompositions of non-convex polygons and an O(nN('3)) heuristics for decomposing three-dimensional polyhedra. The latter procedure is worst-case optimal in the number of convex parts (within a constant multiplicative factor). In both cases, n denotes the total number of vertices, while N refers to the number of edges which exhibit reflex angles. We further explore the problem of finding minimal decompositions in three dimensions and prove its effective decidability. We also establish an (OMEGA)(N('2)) lower bound on the number of convex parts, and use this result to analyze the performance of the above heuristics. The second purpose of this study is to show how convexity can be used for greater efficiency. We justify this claim by studying one of the most fundamental questions in computational geometry: 'Do two convex objects intersect?' Note that the problem does not call for an actual computation of the intersections, which allows the possibility of sub-linear algorithms. The restriction to a simple detection rather than a complete computation is common in many applications areas where efficiency is the main concern. We present a class of practical algorithms for detecting intersections of lines, planes, polygons, and polyhedra in two and three dimensions. Their run-times range from O(log n) for the planar cases to O(log('3)n) for detecting the intersection of two polyhedra, where n represents the total number of vertices involved.

Polygon comparison using a graph representation

Abstract: All of the information necessary to perform the polygon set operations (union, intersection, and difference) and therefore polygon clipping can be generated by a single application of a process called polygon comparison. This process accepts two or more input polygons and generates one or more polygons as output. These output polygons contain unique homogenous areas, each falling within the domain of one or more input polygons. Each output polygon is classified by the list of input polygons in which its area may be found. The union contour of all input is also generated, completing all of the information necessary to perform the polygon set operations.This paper introduces a polygon comparison algorithm which features reduced complexity due to its use of a graph data representation. The paper briefly introduces some of the possible approaches to the general problem of polygon comparison including the polygon set and clipping problems. The new algorithm is then introduced and explained in detail.The algorithm is sufficiently general to compare sets of concave polygons with holes. More than two polygons can be compared at one time; all information for future comparisons of subsets of the original input polygon sets is available from the results of the initial application of the process.The algorithm represents polygons using a graph of the boundaries of the polygons. These graphs are imbedded in a two dimensional geometric space. The use of the graph representation simplifies the comparison process considerably by eliminating many special cases from explicit consideration.Polygon operations like the ones described above are useful in a variety of application areas, especially those which deal with problems involving two dimensional or projected two dimensional geometric areas. Examples include VLSI circuit design, cartographic and demographic applications, and polygon clipping for graphic applications such as viewport clipping, hidden surface and line removal, detailing, and shadowing.

On the extreme rays of the metric cone

Abstract: Introduction. A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equations

Properties of ergodic random mosaic processes

Abstract: The consequences of an ergodic assumption for mosaic processes of random convex polygons are explored in detail. Under certain regularity conditions on the “smallness” and “largeness” of polygons it is shown that the geometric characteristics of the so-called “typical” polygons do in fact exist. New formulae concerning these characteristics are given. The polygon process formed by a Poisson line process is considered as an example of the general theory and, as a result, certain properties of this example which were previously given heuristically, are proved. Edge effects are treated rigorously.

The Curvature of Most Convex Surfaces Vanishes Almost Everywhere

Abstract: At least since 1904 we know about the existence of singular monotone functions, i.e. continuous monotone functions of one real variable with vanishing derivative a.e. In that year Lebesgue [7] and Minkowski [8] gave their famous examples. In 1910 Faber [4], in 1916 Sierpinski [10] and then many others greatly enriched the variety of known examples. By integrating anyone of these functions we get a differentiable convex function with vanishing second derivative a.e. Thus we obtain smooth convex curves with vanishing curvature a.e. A very beautiful and simple example is due to de R h a m [3J: Take a convex polygon, consider on each side the two points dividing it into three equal parts and take the convex polygon with all these division points as vertices. Repeat the procedure. As a limit we get a smooth, strictly convex curve with a.e. vanishing curvature! The purpose of this paper is to show that, in the sense of Baire categories, most convex curves have the above property. More generally, most convex surfaces in IR" have a.e. a vanishing sectional curvature in every tangent direction. We start with some definitions. By a convex surface we shall always understand a closed convex surface in Busemann's sense (see [11, p. 3). A convex curve is a one-dimensional convex surface. A point x of a convex surface S is called smooth if S is differentiable at x. S is smooth if each of its points is smooth. Any half-line in a supporting plane (this is a hyperplane, see [1], p. 4) of a convex surface S, originating at a point x ~ S will be called supporting direction at x. A supporting direction at a smooth point is called tangent direction. The union of a circular with a square 2-cell, such that the radius and centre of the circle coincide with the side-length and a vertex of the square, is called corner-disk. The two points lying on both the circle and the square are called touching points of the corner-disk. A 2-cell having the union of half a circle with a segment as boundary is called semidisk. The points of the boundary of a semidisk which are not smooth are called corners of the semidisk. The radius of a corner-disk or a semidisk is the radius of the circle appearing in their definitions.

Napoleon's Theorem and the Parallelogram Inequality for Affine-Regular Polygons

Abstract: (1980). Napoleon's Theorem and the Parallelogram Inequality for Affine-Regular Polygons. The American Mathematical Monthly: Vol. 87, No. 8, pp. 644-648.

Maximization of lower semi-continuous convex functional on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems

Abstract: We prove, for a proper lower semi-continuous convex functional ƒ on a locally convex space E and a bounded subset G of E, a formula for sup ƒ(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in [7], with the difference that for sup ƒ(G) Lagrange multiplier functionals need not exist. When ƒ is also continuous we give some necessary conditions for g0 ∈ G to satisfy ƒ(g0) = sup ƒ(G). Also, we give some applications to deviations and farthest points. Finally, we show the connections with the “hyperplane theorems” of our previous paper [8].

Computation of the Integral of the Bivariate Normal Distribution Over Convex Polygons

Abstract: A method for computing the integral of the bivariate normal density function over a convex polygon is given. A comparison with two recently published methods is made.

Multiple intersections of diagonals of regular polygons, and related topics

Abstract: Our main concern is to investigate geometrically all sets of three concurrent chords of regular polygons or, equivalently, all adventitious quadrangles (that is, all quadrangles such that the angle between every pair of the six sides is an integral multiple of π/n radians). Most of our results are stated without proof. The proofs are elementary, often consisting of straightforward verification; to include them would make the paper much longer and less readable.

Collapsing three-dimensional convex polyhedra: correction

Abstract: An ingenious construction due to Connelly and Henderson (2) has shown that there exists a rectilinearly triangulated convex polyhedron P in having the property that at least one vertex of the triangulation lies in the interior of a face of P , and yet there is no isomorphic triangulation of a convex polyhedron P ′ all of whose vertices are vertices of P ′. Thus the assertion beginning on the top line of p. 354 of (1) is false, which leaves a gap in the proof of essentially the main result of (1), namely that any rectilinearly triangulated convex polyhedron in can be simplicially collapsed onto its boundary minus a 2-simplex σ. The purpose of this note is to show that the theorem is nevertheless still true. In any case the Corollaries 2 and 3 in (1) are unaffected by the error.

The mean value of the area of polygons circumscribed about a plane convex body

Abstract: Inner parallel bodies are used to prove that the mean area of polygons circumscribed about a convex bodyK of given area is minimum whenK is a circle.

The existence of convex spherical metrics, all closed nonselfintersecting geodesics of which are hyperbolic

Abstract: In this paper it is shown that in any C1-neighborhood of the standard metric H0 on S2, there exists a subset consisting of convex metrics, which is open in the C2-topology, and all of whose closed nonselfintersecting geodesics are hyperbolic.Bibliography: 13 titles.

Synchronous belt drive system

Abstract: A synchronous belt drive system, particularly for small drive pulleys, and method of making such pulleys. The pulley has an optimum pitch radius for its belt teeth addendum circle which has a size between the limits of a maximum pitch radius determined by the radius of a circle through the points of a regular polygon having N equal sides and a minimum pitch radius determined by the radius of a circle having N arcs each equal in arcuate length to the length of each side of said polygon.

A non-convex generalization of the circle problem.

Abstract: §

Die konvexen Polytope im ℝ4, bei denen alle Facetten reguläre Tetraeder sind

Abstract: All Convex Polytopes in ℝ4the Facets of Which Are Regular Tetrahedra. Continuing in ℝn (n≥4) the study of convex polytopes the facets of which are regular, it is proved: Regular polytopes and the two bipyramids over the tetrahedron and the icosahedron are the only convex polytopes in ℝ4 the facets of which are regular tetrahedra.

Lattice panel assembled cage type enclosure structure - has three dimensional polygonal panels joined at edges without supports or frame

Abstract: The space-enclosing configuration, can be used as a hangar-type enclosure for ball games courts, large aviaries or other purposes. It is made up of flat, rigid lattice panels shaped as regular polygons, exclusively made up in turn from intersecting and interconnected lattice bars. The panels (1, 1', 2) are directly joined to each other at their edges, to make up a three-dimensional polygon with no supports or frames. They may form regular pentagons and hexagons e.g. with peripheral bars (5) running along their edges. The whole entity may be set on one or more supports.

Optimization of nested polygons

Abstract: The optimization problem under consideration requires to find the largest regular polygon withk sides to be fitted into a regular polygon withk − 1 sides. If the sequence of these maximal polygons is started with an equilateral triangle, then the final nested polygon, a circle, possesses a radiusr=0.3414r 3, wherer 3 is the radius of the inscribed circle of the equilateral triangle. Lower bounds for the ratior/r 3 are also obtained.

Quasi fircke polygons and the Nielsen convex region

Abstract: It is a classical problem to represent noncompact manifolds by polygons with certain sides identified [1, p. 102]. One attempt has already been made [33. Although the theorem in [3] is false (see [7], the introduction) it could probably be proved for F acting on K(F), the interior of its Nielsen convex region. In this paper we show that any non-elementary fuchsian group F acting on K(F) has a fundamental polygon which is similar to the classical Fricke polygons. It is known [7] that for some F acting on the unit disc A no such polygons exist. The results in this paper will enable us to topologically represent all triangulable surfaces S, considered as a surface without border, by polygons with sides identified. Moreover, these polygons will not have any free sides and the pairs of the identified sides will be organized similar to the classical case. Here one must realize that all triangulable surfaces can be given a conformal structure [1, p. 127].

Easy Determination of the Characteristic Impedance of the Coaxial System Consisting of an Inner Regular Polygon Concentric with an Outer Circle (Short Papers)

Abstract: This paper gives a simple method for the determination of the characteristic impedance of an inner regular polygon concentric with an outer circle. The approach makes use of the method of superposition for plane sheets of charge which were radially disposed in the polygon. The results are in good agreemnt with those obtained by Laura and Luisoni.

Set of mosaic pieces

Abstract: may be formed a mosaic element set that od for use as a game, as a pavement, flooring, tiles, wallpaper patterns, ornamental pattern. like., consists of one another in pairs, different polygonal shaped mosaic elements and can for filling an area defined by a regular polygon having an even page number is limited, or to fill the Euclidean plane are used. The mosaic elements (1 to 16) can be characterized construct that one divides the regular polygon into a set of rhombs (1 to 16b), the plurality of subsets from among themselves equal but contains from subset to subset different shaped diamonds. A first group of mosaic tesserae (1 to 4) takes the form of different among themselves rhombus, and the remaining rhombs (5a to 16b) are in pairs composed so that different mosaic elements (5 to 16) yield, but the sides of the two diamonds in the places where they meet, may not be collinear.