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# Topic: Inversive geometry

###### In the News (Fri 26 Apr 19)

 Inversive geometry -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-13) For two dimensions, conformal geometry with reflections aka inversive geometry is simply the (additional info and facts about Riemann sphere) Riemann sphere with antimeromorphic transformations. Inversive geometry/ (A non-Euclidean geometry in which it is assumed that through any point there are two or more parallel lines that do not intersect a given line in the plane) hyperbolic geometry duality This is usually referred to as the (Euclidean) conformal geometry/ (A non-Euclidean geometry in which it is assumed that through any point there are two or more parallel lines that do not intersect a given line in the plane) hyperbolic geometry duality. www.absoluteastronomy.com /encyclopedia/I/In/Inversive_geometry.htm   (707 words)

 [No title] Inversive geometry is defined by a certain transformation on the Cartesian plane. In inversive geometry, arithmetic is extended to include oo as a num- ber with respect to division: D = oo, 1/Doo = 1/oo = 0, and 1/0 = oo. It is peculiar to inversive geometry because of the way this special case of the BMI is defined: All points must have inverses, including the point at the origin. www.math.yorku.ca /Who/Faculty/Steprans/Courses/3500/Infinity/bmi165-172.txt   (2182 words)

 Teaching Geometry to Artists by J.M. Rees in the Nexus Network Journal vol. 7 no. 1 (Spring 2005)   (Site not responding. Last check: 2007-10-13) Topology is the geometry of continuity, the last in a series of geometries whose definitions of equivalence become progressively more difficult to describe to students with little formal mathematical education. Affine geometry is a special case of projective geometry, where the vanishing point [7] is located at infinity. Inversive geometry is the first non-Euclidean geometry in the sense that it violates Euclid's assumption that parallel lines never meet.[8] Inversive geometry shows the way the interior of a circle is symmetrical to its exterior [Ogilvy 1969]. www.nexusjournal.com /Rees.html   (5353 words)

 Encyclopedia: Circle In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. Jump to: navigation, search The power of a point P with respect to a circle with center C and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. www.nationmaster.com /encyclopedia/Circle   (1506 words)

 Geometry Resources -- All Levels Geometry and the Imagination in Minneapolis - Geometry exercises for a two-week summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991. Geometry Formulas and Facts - Excerpts from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (1995), namely, the geometry section minus differential geometry. Geometry from the Land of the Incas - Presents problems involving circles and triangles, with proofs, SAT practice quizzes and famous quotes. www.edinformatics.com /geometry.htm   (839 words)

 Inversive ring geometry - Wikipedia, the free encyclopedia In mathematics, inversive ring geometry is the extension to the context of associative rings, of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio, concepts usually built upon rings that happen to be fields. In 1968 I.M. Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian, wherein he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Walter Benz developed the commutative ring inversive geometry in the text of 1973 cited above. en.wikipedia.org /wiki/Inversive_ring_geometry   (766 words)

 Inversive geometry - TheBestLinks.com - Circle, Euclidean geometry, Geometry, Mathematics, ...   (Site not responding. Last check: 2007-10-13) Inversive geometry - TheBestLinks.com - Circle, Euclidean geometry, Geometry, Mathematics,... Inversive geometry, Circle, Euclidean geometry, Geometry, Mathematics, Radius... For two dimensions, however, conformal geometry is simply the Riemann sphere. www.thebestlinks.com /Inversive_geometry.html   (298 words)

 algebraic geometry --  Britannica Concise Encyclopedia - Your gateway to all Britannica has to offer!   (Site not responding. Last check: 2007-10-13) The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. He was one of the founders of projective geometry, a branch of mathematics dealing with the relationships between geometric figures and their projected images on a surface or line. concise.britannica.com /ebc/article-9111072   (1053 words)

 AllRefer.com - circle (Mathematics) - Encyclopedia Greek geometry left many unsolved problems about circles, including the problem of squaring the circle, i.e., constructing a square with an area equal to that of a given circle, using only a straight edge and compass; it was finally proved impossible in the late 19th cent. In modern mathematics the circle is the basis for such theories as inversive geometry and certain non-Euclidean geometries. In religion and art it frequently symbolizes heaven, eternity, or the universe. reference.allrefer.com /encyclopedia/C/circle.html   (431 words)

 Geometric structures: History and Development The classical geometry of Euclidean space had developed branches, such as the inversive geometry based on properties of circles, or projective geometry dealing with perspective. In Klein's approach, a geometry is the study of quantities which are unchanged by the transformations preserving the geometry, such as rigid motions in Euclidean geometry, or angle-preserving transformations in conformal geometry. Using the theory of Lie groups as the basis for the infinitesimal geometry (like the quadratic functions defined infinitesimally on Riemannian and Lorentzian manifolds), geometric structures could be defined using extremely general local symmetries. www.math.umd.edu /~caj/EaGLe/history.html   (673 words)

 Definition of Conformal geometry   (Site not responding. Last check: 2007-10-13) In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a "Euclidean-like space with a point added at infinity", or a "Minkowski-like space with a couple of points added at infinity". That is, the setting is a compactification of a familiar space; the geometry is concerned with the implications of preserving angles. For the Euclidean space case, two-dimensional conformal geometry is that of the Riemann sphere. www.pricex.com /definition/Conformal_geometry   (231 words)

 Geometry by David A. Brannan [ISBN: 0521591937] - Find Cheap Textbook Prices & Save BIG During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations. www.gettextbooks.com /isbn_0521591937.html   (2215 words)

 A Science as a Geometry   (Site not responding. Last check: 2007-10-13) Many new geometries were soon invented: non-Euclidean geometry (1826), affine geometry, projective geometry, conformal geometry, inversive geometry, Riemann's geometry (1854), and topological geometry. Klein showed (see section 3.5) that the Euclidean Geometry could be seen as an evolutionary sequence of more and more complex geometry: topological, projective, similarity group, Euclidean geometry, each one being a sub-group of symmetry transformation as the preceding one. This geometry has to be a framework for the expression of the image structural hierarchies. www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node20.html   (1444 words)

 [No title] Geometry and the Imagination in Minneapolis - A collection of handouts during a two-week summer workshop held at the Geometry Center, from June 17-28, 1991. Geometry Page of the Interactive Mathematics Miscellany and Puzzles - Includes various geometry topics, some accompanied by Java-enabled applets, for teachers who want to enliven their web pages or classroom presentation. Geometry Step by Step from the Land of the Incas - A mixture of interactive visuals, sounds and Incan history to stimulate students' interest in Euclidean geometry. botw.org /top/Science/Math/Geometry   (706 words)

 Geometry - Cambridge University Press   (Site not responding. Last check: 2007-10-13) The approach used is that a geometry is a space together with a set of transformations of that space (as argued by Klein in his Erlangen programme). In each case the key results are explained carefully, and the relationships between the geometries are discussed. This richly illustrated and clearly written text includes full solutions to over 200 problems, and is suitable both for undergraduate courses on geometry and as a resource for self study. www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521597870   (191 words)

 Hyperbolic Geometry   (Site not responding. Last check: 2007-10-13) Parts of the tessellation are shown in varying degrees of layers in each of the quarters. It was with this in mind that I started to construct a series of Cabri macros and an ever growing menu for constructions for hyperbolic (or non-Euclidean) geometry in the Poincaré disc model. The menu commands can be used to draw figures that illustrate some of the fascinating results and figures to be found in the hyperbolic plane. mcs.open.ac.uk /tcl2/nonE/nonE.html   (353 words)

 Science Math Geometry   (Site not responding. Last check: 2007-10-13) Geometry exercises for a two-week summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991. Geometry professors and instructors may keep informed of the latest research conducted in their field at the geometry-research discussion list. Cabri constructions for the demonstration of the basic concepts of hyperbolic geometry in the Poincare disc model. www.allin1sports.com /dir.cfm?cat=Science/Math/Geometry   (712 words)

 Amazon.com: Geometry Revisited (New Mathematical Library): Books: H. S. M. Coxeter,Samuel L. Greitzer   (Site not responding. Last check: 2007-10-13) inversive distance, cyclic quadrangle, orthic triangle, spiral similarity, pedal triangle, central dilatation, coaxal circles, three cevians, medial triangle, radical axis, inversive plane, radical axes, central conic, external bisectors, internal bisectors, circles orthogonal, three collinear points, congruent circles, inverse points, two intersecting circles, circle with center, angle bisectors, directed segments, direct similarity, cross ratio A thorough knowledge of geometry beyond what you learn in high school in necessary needed to be competitive, and the place to get it is Geometry Revisited. Geometry Revisited should be used as a textbook for every geometry class in high school. www.amazon.com /exec/obidos/tg/detail/-/0883856190?v=glance   (1232 words)

 Inverse - Wikipedia, the free encyclopedia Something that is inverted is something that is flipped over, around or otherwise appearing in an opposite manner than is normal, customary, or common. An inverted river delta is a river delta that has an mirror-imaged geometry compared to normal river deltas. Inverting an object is often referred to flipping it upside down. en.wikipedia.org /wiki/Inversion   (285 words)

 Geometry Theorem 1 The locus of all points whose powers with respect to two nonconcentric circles are equal is a line perpendicular to the line of centers of the two circles. For Eucleadean geometry the important transformations are isometries. Similarly for definition of the inversive plane we could make an extension of Euclidean plane for the projective case. www.amsta.leeds.ac.uk /~kisilv/courses/math255.html   (4657 words)

 The Geometry Junkyard: All Topics This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube. The Geometry Center's collection includes programs for generating Penrose tilings, making periodic drawings a la Escher in the Euclidean and hyperbolic planes, playing pinball in negatively curved spaces, viewing 3d objects, exploring the space of angle geometries, and visualizing Riemann surfaces. Geometry forum discussion on the Reuleux triangle and its ability to drill out (most of) a square hole. www.ics.uci.edu /~eppstein/junkyard/all.html   (9742 words)

 The Cyber Murid Open Directory   (Site not responding. Last check: 2007-10-13) Geometry exercises for a twoweek summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991. Search the Geometry Problem of the Week's archive of creative, nonroutine challenges, as well as submissions and commentary, dating back to 1993. Geometry professors and instructors may keep informed of the latest research conducted in their field at the geometryresearch discussion list. www.cybermurid.com /bin/odp/index.cgi?base=/Science/Math/Geometry   (704 words)

 Science - Math - Geometry - Newsletter - News - Reviews - Education - Ratings Hyperbolic Geometry This page and links maintained by Tim Lister, t.c.lister@open.ac.uk Last updated: 24/09/01 A tessellation of the hyperbolic plane H2 (the Poincar unit disc model) by (2, i, i) triangles, that is, with angles (90, 0, 0 Every triangle has two... Geometry Formulas and Facts This document is excerpted from the 30th Edition of the 160; CRC Standard Mathematical Tables and Formulas, published in late 1995 by CRC press. Fundamentals of Geometry Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry While there are no royal roads to geometry, I invite you on a hitchhiking trip... www.banner-net.com /Science/Math/Geometry   (2101 words)

 [No title]   (Site not responding. Last check: 2007-10-13) Freudenthal, Hans "The Main Trends in the Foundations of Geometry in the 19th Century." Logic, Methodology and Philosophy of Science, ed. It includes Euclidean, Hyperbolic and Projective Geometry.} Richards, Joan L., Mathematical Visions: the Pursuit of Geometry in Victorian England, Academic Press, 1988 (QA443.5.R53 1988). { A study of the progressive acceptance of non-Euclidean geometry in England through the 19th century. The last chapter describes the generalization to Minkowskian geometry, which is the geometry of relativistics mechanics.} Yaglom, I.M., Felix Klein and Sophus Lie, Birkhauser,Boston, 1988. www.math.carleton.ca:16080 /~mortimer/courses/Web4207/Administration/4207-Bibliography.doc   (1379 words)

 Directorio - Enlacesfinancieros.com/directorio   (Site not responding. Last check: 2007-10-13) 'Educational Geometry' is based on visual geometry, linked to philosophy and visual art and the learning is free of the application of formulae or the use of computations. Metric geometry section of the mathematics e-print arXiv. Symplectic geometry section of the mathematics e-print arXiv. www.enlacesfinancieros.com /directorio/directory.cgi?dir=/Science/Math/Geometry   (676 words)

 Round Triangles In inversive geometry, a triangle ABC is bounded by three arcs of circles: AB, BC, and CA. When this point P happens to be the point at infinity of the inversive plane, then triangle ABC is a Euclidean triangle with straight lines as sides. There are many very nice properties of Euclidean triangles, and these "polar triangles" in inversive geometry enjoy the same properties. aleph0.clarku.edu /~djoyce/java/round/round1.html   (564 words)

 Ellipses in the Inversive Plane - Coffman, Frantz (ResearchIndex) 4 The invariant theory of the inversion group: geometry upon a.. 2 erential geometry of plane curves (context) - Cairns, Sharpe et al. 2 erential invariants of inversive geometry (context) - Patterson, di - 1928 citeseer.ist.psu.edu /540201.html   (662 words)

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