Showing papers on "Regular polygon published in 1992"

Lattice translates of a polytope and the Frobenius problem

Abstract: This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2,...an such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the “covering radius”. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytopeK, the setK+ℤh={x∶x∈ℝn;x=y+z;y∈K;z∈ℤn} which is the portion of space covered by all lattice translates ofK. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovasz [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovasz and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.

Computing the antipenumbra of an area light source

Abstract: We define the antiumbra and the antipenumbra of a convex areal light source shining through a sequence of convex areal holes in three dimensions. The antiumbra is the volume beyond the plane of the final hole from which all points on the light source can be seen. The antipenumbra is the volume from which some, but not all, of the light source can be seen. We show that the antipenumbra is, in general, a disconnected set bounded by portions of quadric surfaces, and describe an implemented O(n2) time algorithm that computes this boundary. The antipenumbra computation is motivated by visibility computations and might prove useful in rendering shadowed objects. We also present an implemented extension of the algorithm that computes planar and quadratic surfaces of discontinuous illumination useful for polygon meshing in global illumination computations.

An Equal-Area Map Projection For Polyhedral Globes

Abstract: Numerous polyhedral shapes have been proposed as approximations for globes, and the projection most often used is the Gnomonic, with considerable scale and area distortion. Complicated conformal projections have been designed, but an equal-area projection has been used only once, for the icosahedron. The Lambert Azimuthal Equal-Area projection can be modified to provide an exactly fitting, perfectly equal-area projection for any polyhedral globe that has regular polygons, but is most satisfactory for the dodecahedron with 12 pentagons and for the truncated icosahedron with 20 hexagons and 12 pentagons. On the application to the truncated icosahedron, the angular deformation does not exceed 3.75°, and the scale variation is less than 3.3 percent. These advantages are at the expense of increased interruptions at the polygon edges when the polyhedral globe is unfolded. On a propose de nombreuses formes polyedriques comme approximation de globes et la projection gnomonique est la plus souvent utilisee, avec d...

Convex decomposition of polyhedra and robustness

Abstract: This paper presents a simple algorithm to compute a convex decomposition of a nonconvex polyhedron of arbitrary genus (handles) and shells (internal voids) For such a polyhedron S with n edges and rnotches (features causing nonconvexity in polyhedra), the algorithm produces a worst-case optimal $O(r^2 )$ number of convex polyhedra $S_i $, with $U_{i = 1}^k S_i = S$, in $O(nr^2 + r^{7/2} )$ time and $O(nr + r^{5/2} )$ space Recently, Chazelle and Palios have given a fast $O((n + r^2 )\log r$) time and $O(n + r^2 )$ space algorithm to tetrahedralize a nonconvex polyhedron Their algorithm, however, works for a simple polyhedron of genus zero and with no shells (internal voids) The algorithm, presented here, is based on the simple cut and split paradigm of Chazelle With the help of zone theorems on arrangements, it is shown that this cut and split method is quite efficient The algorithm is extended to work for a certain class of nonmanifold polyhedra Also presented is an algorithm for the same problem

The Pentagram Map

Abstract: We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagon's diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a “pentagram map” on convex polygons of more than five sides, showing that iterated images of anyinitial polygon converge exponentially fast to a point. We conjecture that the asymptotic behavior of this convergence is the same as under a projective transformation. Finally, we show a connection between the pentagram map and a certain flow defined on parametrized curves.

Inner and outer j -radii of convex bodies in finite-dimensional normed spaces

Abstract: This paper is concerned with the various inner and outer radii of a convex bodyC in ad-dimensional normed space. The innerj-radiusrj(C) is the radius of a largestj-ball contained inC, and the outerj-radiusRj(C) measures how wellC can be approximated, in a minimax sense, by a (d --j)-flat. In particular,rd(C) andRd(C) are the usual inradius and circumradius ofC, while 2r1(C) and 2R1(C) areC's diameter and width. Motivation for the computation of polytope radii has arisen from problems in computer science and mathematical programming. The radii of polytopes are studied in [GK1] and [GK2] from the viewpoint of the theory of computational complexity. This present paper establishes the basic geometric and algebraic properties of radii that are needed in that study.

Delaunay's mesh of a convex polyhedron in dimension d. application to arbitrary polyhedra

Abstract: This paper presents a method for creating a Delaunay triangulation connected to a set of specified points. The theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach, valid for dimensions 2 and 3, which is simple, robust and well adapted to computation. Convex polyhedral and arbitrary polyhedral situations are introduced.

A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere

Abstract: We describe a characterization of convex polyhedra in $\h^3$ in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in $\E^3$ all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832.

Finding convex edge groupings in an image

Abstract: In an image, there are groups of intensity edges that are likely to have resulted from the same convex object in a scene. A new method for identifying such groups is described here. Groups of edges that form a convex polygonal chain, such as a convex polygon or a spiral, are extracted from a set of image edge fragments. A key property of the method is that its output is no more complex than the original image. The method uses a triangulation of the linear edge segments in an image to define a local neighborhood that is scale invariant. From this local neighborhood a local convexity graph is constructed; this encodes which neighboring image edges could be part of a convex group. A path in the graph corresponds to a convex polygonal chain in the image, with a cyclic path corresponding to a polygon. We have implemented the method and found that it is efficient in practice as well as in theory. Examples are presented to illustrate that the technique finds intuitively salient groups, including for images of cluttered scenes.

On the Number of Convex Lattice Polygons

Abstract: Note: Professor Pach's number: [093] Reference DCG-ARTICLE-2008-017doi:10.1017/S0963548300000341 Record created on 2008-11-17, modified on 2017-05-12

Computation of certain measures of proximity between convex polytopes: a complexity viewpoint

Abstract: The quantification of proximity between a pair of objects whose point descriptions are given is considered. Four problems of proximity between two convex polytopes in R/sup 3/ are considered. The convex polytopes are represented as convex hulls of finite sets of points. The authors discuss the complexity of solving the four problems. They analyze algorithms for the four problems in terms of two complexity types. Let the total number of points in the two finite sets be n. It is shown that three of the proximity problems, checking intersection, checking whether the polytopes are just touching, and finding the distance between them, can be solved in O(n) time for fixed s and in polynomial time for varying s. It is also shown that the fourth proximity problem of finding the intensity of collision for varying s is NP-complete. >

Triangulating polygons without large angles

Abstract: We show how to triangulate an n-vertex polygonal region—adding extra vertices as necessary—with triangles of guaranteed quality. Using only O(n) triangles, we can guarantee that the smallest height (shortest dimension) of a triangle is approximately as large as possible. Using O(n log n) triangles, we can also guarantee that the largest angle is no greater than 150°. Finally we give a nonobtuse triangulation algorithm for convex polygons that uses O(n1.85) triangles.

Real and Complex Asymptotic Symmetries in Quantum Gravity, Irreducible Representations, Polygons, Polyhedra, and the A, D, E Series

Abstract: The Bondi-Metzner-Sachs group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of general relativity. However, in quantum gravity, complexified or euclidean versions of general relativity are frequently considered, and the question arises: Are there similar symmetry groups for these versions of the theory? In this paper it is shown that there are such analogues of B, and a variety of further ones, either real in any signature, or complex. The relationships between these various groups are described. Irreducible unitary representations (IRS) of the complexification [Note: See the image of page 271 for this formatted text] CB of B itself are analysed. It is proved that all induced IRS of [Note: See the image of page 271 for this formatted text] CB arise from IRS of compact \`little groups'. It follows that some IRS of [Note: See the image of page 271 for this formatted text] CB are controlled by the IRS of the \`A, D, E' series of finite symmetry groups of regular polygons and polyhedra in ordinary euclidean 3-space. Possible applications to quantum gravity are indicated.

Alternative proof of sine's theorem on the size of a regular polygon in Rn with the ℓź-metric

Abstract: It is shown that a regular polygon inR n with the (2n) n -metric has at most (2n) n vertices.

Separation and approximation of polyhedral objects

Abstract: Given a family of disjoint polygons P1, P2,…, Pk in the plane, and an integer parameter m, it is NP-complete to decide if the Pi's can be separated by a polygonal family consisting of m edges, that is, if there exist polygons R1, R2,…, Rk with pairwise-disjoint boundaries such that Pi *** Ri and S|Ri| ≤ m. In three dimensions, the problem of separating even two nested convex polyhedra by a k-facet polyhedron is NP-complete. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present efficient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we can separate a convex polyhedron from a nonconvex polyhedron with a convex polyhedral surface whose facet-complexity is O(log n, times the optimal, where n = |P|+|Q| is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O (n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d ≥ 4, the facet-complexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1)time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by non-polyhedral surfaces, such as spherical patches.

X-rays of polygons

Abstract: Various results are given concerning X-rays of polygons in ?2. It is shown that no finite set of X-rays determines every star-shaped polygon, partially answering a question of S. Skiena. For anyn, there are simple polygons which cannot be verified by any set ofn X-rays. Convex polygons are uniquely determined by X-rays at any two points. Finally, it is proved that given a convex polygon, certain sets of three X-rays will distinguish it from other Lebesgue measurable sets.

Stability of a regular polygon of finite vortices

Abstract: It is well known that a system of N point vortices arranged in a circular row, so that the vortices are at the vertices of a regular polygon, is stable if N N = 7 and unstable if N > 7 (Havelock 1931). The effect on this result of taking account of the finite size of the vortices is considered analytically. The vortices are considered to be uniform with small but finite core. Approximate equations for the shape and motion of a vortex subjected to an external velocity field are given and used to evaluate the shape and angular velocity of rotation of the system and to study its stability to plane infinitesimal disturbances. It is found that the system is stable if N N ≥ 7. These asymptotic results for small core area are in general consistent with Dritschel (1985) where the motion and stability of up to N = 8 finite vortices is evaluated numerically; the steady configuration and the stability results for these values of N are in agreement except in a region of parameter space where a high degree of accuracy is required in the numerical calculation to resolve the growth rate of small disturbances. The case of a linear array of finite vortices is obtained as a special limiting case of the system. The growth rate of plane infinitesimal disturbances for this case is given.

Vector-space solution for a morphological shape-decomposition problem

Abstract: We define a restricted domain as the discrete set of points representing any convex, four-connected, filled polygon whose (i) vertices lie on the lattice points, (ii) interior angles are multiples of 45°, and (iii) number of sides are at most eight. We describe the boundary code and discrete half-plane representation and use them for representing restricted domains. Morphological operations of dilation and n-fold dilation on the restricted domains with structuring elements that are also restricted domains are expressed in terms of the above representations. We give algorithms for these operations and prove that they are of O(1) complexity and hence are independent of the size of the objects.

Unit distances between vertices of a convex polygon

Abstract: Many years ago Danzer resolved an open question of Erdos by constructing a convex 9-gon, each vertex of which has the same distance to three other vertices. In Danzer's example, the replicated distance is not the same for all vertices. The present paper shows that it can be the same when n is somewhat larger than 9. In particular, there are convex n-gons with the following property. The vertices are partitioned into sets A and B on opposite sides of a line such that each a ϵ A is distance 1 from three vertices in B and each b ϵ B is distance 1 from three vertices in A. The smallest n for which this is possible is n = 20.

Model-based probing strategies for convex polygons

Abstract: We prove that n+4 finger probes are sufficient to determine the shape of a convex n-gon from a finite collection of models, improving the previous result of 2n+1. Further, we show that n−1 are necessary, proving this is optimal to within an additive constant. For line probes, we show that 2n+4 probes are sufficient and 2n−3 necessary. The difference between these results is particularly interesting in light of the duality relationship between finger and line probes.

Newell's method for computing the plane equation of a polygon

Abstract: Publisher Summary This chapter describes a numerically robust way of computing the plane equation of an arbitrary 3-D polygon, known as Newell's method This technique works for concave polygons and polygons containing collinear vertices as well as for nonplanar polygons—for example, polygons resulting from perturbed vertex locations It can be shown that the areas of the projections of a polygon onto the Cartesian planes—xy, yz, and zx—are proportional to the coefficients of the normal vector to the polygon Newell's method computes each one of those projected areas as the sum of the signed areas of the trapezoidal regions enclosed between each polygon edge and its projection onto one of the Cartesian axes Newell's method may seem inefficient for planar polygons because it uses all the vertices of a polygon when, in fact, only three points are needed to define a plane

Device for microscanning and infrared camera provided with such device

Abstract: The invention relates to a microscanning device for deviating periodically a convergent beam. The device according to the invention is comprised of an assembly (33) consisting of an even number of thin blades (21-28) which are opposite by pairs, and forming a regular polygon. The assembly has a symmetry axis (Δ), arranged perpendicularly to the direction (Rm) of the convergent beam (FC), and is actuated with a uniform rotation motion about the axis (Δ). The invention applies to the microscanning in one or two directions, according to the cross-sectional shape of the thin blades, for bidimensional sensors of infrared cameras.