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Regular polygon

About: Regular polygon is a research topic. Over the lifetime, 4495 publications have been published within this topic receiving 68009 citations.


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TL;DR: In this article, the Grassmann manifold of nonoriented (respectively, oriented) k-planes in R passing through O ∈ R has been studied, where the fiber over an oriented k-plane α ∈ Gk(R) is α itself regarded as a k-dimensional vector space.
Abstract: Topological means are used for the study of approximation of 2-dimensional sections of a 3-dimensional convex body by affine-regular pentagons and approximation of a centrally symmetric convex body by a prism. Also, the problem of estimating the relative surface area of the sphere in a normed 3-space, the problem on universal covers for sets of unit diameter in Euclidean space, and some related questions are considered. Throughout, by a convex body K ⊂ R (a figure for n = 2) we mean a compact convex subset of R with nonempty interior. We denote by Gk(R) (respectively, G+k (R )) the Grassmann manifold of nonoriented (respectively, oriented) k-planes in R passing through O ∈ R. We let γ k : Ek(R ) → Gk(R) and (γ k ) : E k (R) → G + k (R ) be the tautological fiber bundles, where the fiber over an (oriented) k-plane α ∈ Gk(R) is α itself regarded as a k-dimensional vector space. We say that a field of convex bodies (or figures ; a CB or a CF-field) is given in a vector bundle γ if in each fiber α of γ we mark a convex body K(α) depending continuously on α. A CB-field is pointed if for each α we also mark a point x(α) ∈ K(α) depending continuously on α. (In other words, x(α) is a section of γ.) If λ ∈ R and P is an affine-regular polygon (e.g., a pentagon or a parallelogram) with center O(P ), then λP denotes the polygon homothetic to P with homothety ratio λ and homothety center O(P ). We denote by S(K) the area of a figure K ⊂ R. §1. Fields of convex figures in γ 2 and (γ 2), and 2-dimensional sections of convex bodies in R First, we prove two corollaries to the following known result. Theorem [2]. Each CF-field in γ 2 contains a figure circumscribed about an affine-regular octagon. Corollary 1. Suppose C is the bounded component of a cubic surface in R. Then each inner point O of C lies in a plane intersecting C along an ellipse. Proof. Indeed, C is convex automatically, because otherwise C intersects some line at 4 points. Consequently, the section of C by some 2-plane through O is circumscribed about an affine-regular octagon. Then this section is a component of a cubic, intersects an ellipse at 8 points, and, consequently, is an ellipse by the Bézout theorem. 2000 Mathematics Subject Classification. Primary 52A10, 52A15.

1 citations

Journal ArticleDOI
TL;DR: In this paper, an upper bound on the number of edge-to-edge, irreducible decompositions of a centrally symmetric convex (2k)-gon into two-dimensional convex pieces was proved.
Abstract: In this paper we deal with edge-to-edge, irreducible decompositions of a centrally symmetric convex (2k)-gon into centrally symmetric convex pieces. We prove an upper bound on the number of these decompositions for any value of k, and characterize them for octagons.

1 citations

DOI
01 Jun 2019
TL;DR: In this article, a regular polygon with $n$ sides is described by a periodic (circular) sequence with period n$ and each element of the sequence represents a vertex of the polygon.
Abstract: ‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and generates a processed circular output sequence‎. ‎The system can be described by a permutation function‎. ‎Permutation functions can be written in a simple form using circular indexation‎. ‎The operation between the symmetries of the polygon is reduced to the composition of permutation functions‎, ‎which in turn is easily implemented using periodic sequences‎. ‎It is also shown that each symmetry is effectively a pure rotation or a pure flip‎. ‎It is also explained how to synthesize each symmetry using two generating symmetries‎: ‎time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by $2pi‎ /‎n$ radians clockwise)‎. ‎The group of the symmetries of a polygon is called a dihedral group and it has applications in different engineering fields including image processing‎, ‎error correction codes in telecommunication engineering‎, ‎remote sensing, and radar‎.

1 citations

Patent
James Acquavella1
13 Apr 2007
TL;DR: In this article, a star shape with a number of star points equal to an integer portion of the star points input parameter plus one is rendered as regular polygon sides and two sides are rendered as adjacent symmetrical bezier curves, and the modified polygon shape is stored or rendered to an output device.
Abstract: Methods and apparatus disclosed herein receive a non-integer star points input parameter value, create a star shape with a number of star points equal to an integer portion of the star points input parameter plus one, and store or render the star shape to an output device. A fractional star point is differentiated from integer star points by a differential radial length between the two. Some embodiments also receive a non-integer polygon sides input parameter value and render a modified polygon shape with a total number of sides equal to an integer portion of the polygon sides input parameter plus one. A number of sides equal to the integer portion of the polygon sides parameter minus one are rendered as regular polygon sides. Two sides are rendered as adjacent symmetrical bezier curves, and the modified polygon shape is stored or rendered to an output device.

1 citations

Posted Content
TL;DR: It is proved that every such grid contains a convex polygon with Ω(log n) vertices and that this bound is tight up to a constant factor, and a tight lower bound is obtained for the maximum number of points in convex position in a d-dimensional grid.
Abstract: We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers. First, we prove that every such grid contains a convex polygon with $\Omega$(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d $\in$ N), and obtain a tight lower bound of $\Omega$(log d--1 n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the largest convex chain in a grid that contains no two points of the same x-or y-coordinate. We show how to efficiently approximate the maximum size of a supported convex polygon up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

1 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
20231,380
20222,904
2021399
2020310
2019214