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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 calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trott Bitangents
Trott may refer to: * Abby Trott, American voice actress * Albert Trott (1873–1914), Australian-born Test cricketer for both Australia and England * Benjamin Trott (born 1977), American blogger and businessman * Christopher Trott (born 1988), English YouTuber and musician * Dave Trott, American Congressional representative from Michigan * Emma Trott (born 1989), British road and track cyclist * Harry Trott (1866–1917), Australian Test cricketer * Jonathan Trott (born 1981), English cricketer *Josephine Trott (1874–1950), composer * Laura Trott (born 1992), British track cyclist * Mena Grabowski Trott (born 1977), American blogger and businesswoman * Novella Jewell Trott (1846–1929), American author and editor * Stephen S. Trott (born 1939), American judge * Stuart Trott Stuart Trott (born 25 April 1948) is a former Australian rules footballer in the Victorian Football League. He came from Frankston originally and is the great grandson of 1896 Test cricket captain Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Shortest Path
The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. Two dimensions In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. These algorithms are based on two different principles, either performing a shortest path algorithm such as Dijkstra's algorithm on a visibility graph derived from the obstacles or (in an approach called the ''continuous Dijkstra'' method) propagating a wavefront from one of the points until it meets the other. Higher dimensions In three (and higher) dimensions the problem is NP-hard in the general case, J. Canny and J. H. Reif,New lower bound techniques for robot motion planning pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 system of publication. Since 2015 the conference proceedings have been published by the Leibniz International Proceedings in Informatics Dagstuhl is a computer science research center in Germany, located in and named after a district of the town of Wadern, Merzig-Wadern, Saarland. Location Following the model of the mathematical center at Oberwolfach, the center is installed in ... 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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 |
Monge's Theorem
In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them. There are two such external tangent lines for any two circles. Each such pair has a unique intersection point in the extended Euclidean plane. Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external tangents. In other words, if the two external tangents are considered to intersect at the point at infinity, then the other two inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, :\,t_^2=\overline^2-(R_i-R_j)^2. We will try to express this length in terms of the points \,K_i,K_j. By the law of cosines in triangle \,O_iOO_j, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 find \varp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous ''Journal'' (1826). After Steiner's publication (1832) of his ''Systematische Entwickelungen'' he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834). This he occupied until his death in Bern on 1 April 1863. He was described by Thomas Hirst as follows: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Lines To Two Circles
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a ''tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
