Topic
Polygon
About: Polygon is a research topic. Over the lifetime, 12552 publications have been published within this topic receiving 173923 citations. The topic is also known as: 2-polytope.
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23 Nov 2012TL;DR: In this article, the authors introduce the Matrix Integrals method and the Goulden-Jackson Formula for computing polygonal gluings, and apply it to the problem of "Mirror Symmetry in Dimension One".
Abstract: 0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.- 5 Meromorphic Functions and Embedded Graphs.- 6 Algebraic Structures Associated with Embedded Graphs.- A.1 Representation Theory of Finite Groups.- A.1.1 Irreducible Representations and Characters.- A.1.2 Examples.- A.1.3 Frobenius's Formula.- A.2 Applications.- A.2.2 Examples.- A.2.3 First Application: Enumeration of Polygon Gluings.- A.2.4 Second Application: the Goulden-Jackson Formula.- A.2.5 Third Application: "Mirror Symmetry" in Dimension One.- References.
800 citations
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
TL;DR: The determinant method, based on the determinant of the covariance matrix of the capture points, is introduced and it is shown that this method, under specified assumptions, is free from both forms of bias.
728 citations
01 Aug 1996
TL;DR: An algorithm for real-time level of detail reduction and display of high-complexity polygonal surface data that has been implemented for approximating and rendering digital terrain models and other height fields, and consistently performs at interactive frame rates with high image quality.
Abstract: We present an algorithm for real-time level of detail reduction and display of high-complexity polygonal surface data. The algorithm uses a compact and efficient regular grid representation, and employs a variable screen-space threshold to bound the maximum error of the projected image. A coarse level of simplification is performed to select discrete levels of detail for blocks of the surface mesh, followed by further simplification through repolygonalization in which individual mesh vertices are considered for removal. These steps compute and generate the appropriate level of detail dynamically in real-time, minimizing the number of rendered polygons and allowing for smooth changes in resolution across areas of the surface. The algorithm has been implemented for approximating and rendering digital terrain models and other height fields, and consistently performs at interactive frame rates with high image quality.
714 citations
TL;DR: The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained.
Abstract: A procedure for computing shaded pictures of curved surfaces is presented. The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained. In order to achieve speed efficiency, the technique developed by Watkins is used which makes possible a hardware implementation of this algorithm.
661 citations