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Bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at . Bitangents of algebraic curves In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Bitangents of polygons The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other. This bitangent ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bitangents Of A Quartic
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane. An explicit quartic with twenty-eight real bitangents was first given by As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines. gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane. Example The Trott curve, another curve with 28 real bitan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Malfatti Circles
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle. Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle. A simple construction of the Malfatti circles was given by , and many mathematicians have since studied the problem. Malfatti himself supplied a formula for the radii of the three circles, and they may also be used to define two triangle centers, the Ajima–Malfatti points of a triangle. The problem of maximizing the total area of three circles in a triangle is never solved by the Malfatti circles. Instead, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Visibility Graph
In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge represents a visible connection between them. That is, if the line segment connecting two locations does not pass through any obstacle, an edge is drawn between them in the graph. When the set of locations lies in a line, this can be understood as an ordered series. Visibility graphs have therefore been extended to the realm of time series analysis. Applications Visibility graphs may be used to find Euclidean shortest paths among a set of polygonal obstacles in the plane: the shortest path between two obstacles follows straight line segments except at the vertices of the obstacles, where it may turn, so the Euclidean shortest path is the shortest path in a visibility graph that has as its nodes the start and destination points and the ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Belt Problem
The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with radius ''r''1 and ''r''2 whose centers are separated by a distance ''P''. The solution of the belt problem requires trigonometry and the concepts of the bitangent line, the vertical angle, and congruent angles. Solution Clearly triangles ACO and ADO are congruent right angled triangles, as are triangles BEO and BFO. In addition, triangles ACO and BEO are similar. Therefore angles CAO, DAO, EBO and FBO are all equal. Denoting this angle by \varphi (denominated in radians), the length of the belt is :CO + DO + EO + FO + \text CD + \text EF \,\! :=2r_1\tan(\varphi) + 2r_2\tan(\varphi) + (2\pi-2\varphi)r_1 + (2\pi-2\varphi)r_2 \,\! :=2(r_1+r_2)(\tan(\varphi) + \pi- \varphi) \,\! This exploits the convenience of denominating angles in radians that the length of an arc = the radius × the measure of the angle facing the arc. To fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pseudotriangulation
In Euclidean plane geometry, a pseudotriangle (''pseudo-triangle'') is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation (''pseudo-triangulations'') is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π. Although the words "pseudotriangle" and "pseudotriangulation" have been used with various meanings in mathematics for much longer, the terms as used here were introduced in 1993 by Michel Pocchiola and Gert Vegter in connection with the computation of visibility relations and bitangents among convex obstacles in the plane. Pointed pseudotriangulations were first considered by Ileana Streinu (2000, 2005) as part of her solution to the carpenter's ruler problem, a proof that any simple polygonal path in the plane can be straightened out by a sequence of continuous motions. P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Casey's Theorem
In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. Formulation of the theorem Let \,O be a circle of radius \,R. Let \,O_1, O_2, O_3, O_4 be (in that order) four non-intersecting circles that lie inside \,O and tangent to it. Denote by \,t_ the length of the exterior common bitangent of the circles \,O_i, O_j. Then: :\,t_ \cdot t_+t_ \cdot t_=t_\cdot t_. Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem. Proof The following proof is attributable to Zacharias. Denote the radius of circle \,O_i by \,R_i and its tangency point with the circle \,O by \,K_i. We will use the notation \,O, O_i for the centers of the circles. Note that from Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Symmetry Set
In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis, a subset of the symmetry set is a set of curves which roughly run along the middle of an object. In 2 dimensions Let I \subseteq \mathbb be an open interval, and \gamma : I \to \mathbb^2 be a parametrisation of a smooth plane curve. The symmetry set of \gamma (I) \subset \mathbb^2 is defined to be the closure of the set of centres of circles tangent to the curve at at least two distinct points (bitangent circles). The symmetry set will have endpoints corresponding to vertices of the curve. Such points will lie at cusp of the evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dijkstra's Algorithm
Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. The algorithm exists in many variants. Dijkstra's original algorithm found the shortest path between two given nodes, but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree. For a given source node in the graph, the algorithm finds the shortest path between that node and every other. It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discrete And Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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International Journal Of Computational Geometry And Applications
The ''International Journal of Computational Geometry and Applications'' (IJCGA) is a bimonthly journal published since 1991, by World Scientific. It covers the application of computational geometry in design and analysis of algorithms, focusing on problems arising in various fields of science and engineering such as computer-aided geometry design (CAGD), operations research, and others. The current editors-in-chief are D.-T. Lee of the Institute of Information Science in Taiwan, and Joseph S. B. Mitchell from the Department of Applied Mathematics and Statistics in the State University of New York at Stony Brook. Abstracting and indexing * Current Contents/Engineering, Computing & Technology * ISI Alerting Services * Science Citation Index Expanded (also known as SciSearch) * CompuMath Citation Index * Mathematical Reviews * INSPEC * DBLP Bibliography Server * Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Symposium On Computational Geometry
The International Symposium on Computational Geometry (SoCG) is an academic conference in computational geometry. It was founded in 1985, and was originally sponsored by the SIGACT and SIGGRAPH Special Interest Groups of the Association for Computing Machinery (ACM). It dissociated from the ACM in 2014, motivated by the difficulties of organizing ACM conferences outside the United States and by the possibility of turning to an open-access Open access (OA) is a set of principles and a range of practices through which research outputs are distributed online, free of access charges or other barriers. With open access strictly defined (according to the 2001 definition), or libre o ... system of publication. Since 2015 the conference proceedings have been published by the Leibniz International Proceedings in Informatics instead of by the ACM. Since 2019 the conference has been organized under the auspices of the newly-formed Society for Computational Geometry. A 2010 assessment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Journal Of Computer And System Sciences
The ''Journal of Computer and System Sciences'' (JCSS) is a peer-reviewed scientific journal in the field of computer science. ''JCSS'' is published by Elsevier, and it was started in 1967. Many influential scientific articles have been published in ''JCSS''; these include five papers that have won the Gödel Prize. an 2014 Gödel Prize Its managing editor is |