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The Real Projective Plane

The Real Projective Plane Author H.S.M. Coxeter
ISBN-10 9781461227342
Release 2012-12-06
Pages 227
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Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (§3.34). This makes the logi cal development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument. H.S.M.C. November 1992 v Preface to the Second Edition Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the prop erties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to non· Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity.

Projective Geometry and Algebraic Structures

Projective Geometry and Algebraic Structures Author R. J. Mihalek
ISBN-10 9781483265209
Release 2014-05-10
Pages 232
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Projective Geometry and Algebraic Structures focuses on the relationship of geometry and algebra, including affine and projective planes, isomorphism, and system of real numbers. The book first elaborates on euclidean, projective, and affine planes, including axioms for a projective plane, algebraic incidence bases, and self-dual axioms. The text then ponders on affine and projective planes, theorems of Desargues and Pappus, and coordination. Topics include algebraic systems and incidence bases, coordinatization theorem, finite projective planes, coordinates, deletion subgeometries, imbedding theorem, and isomorphism. The publication examines projectivities, harmonic quadruples, real projective plane, and projective spaces. Discussions focus on subspaces and dimension, intervals and complements, dual spaces, axioms for a projective space, ordered fields, completeness and the real numbers, real projective plane, and harmonic quadruples. The manuscript is a dependable reference for students and researchers interested in projective planes, system of real numbers, isomorphism, and subspaces and dimensions.

Models of the Real Projective Plane

Models of the Real Projective Plane Author Francois Apery
ISBN-10 9783322895691
Release 2013-03-09
Pages 156
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In the present time, objects generated by computers are replacing models made from wood, wire, and plaster. It is interesting to see how computer graphics can help us to understand the geometry of surfaces and illustrate some recent results on representations of the real projective plane.

Perspectives on Projective Geometry

Perspectives on Projective Geometry Author Jürgen Richter-Gebert
ISBN-10 3642172865
Release 2011-02-04
Pages 571
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Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. In particular, it explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay between geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the author’s experience in implementing geometric software and includes hundreds of high-quality illustrations.

Linear Algebra and Projective Geometry

Linear Algebra and Projective Geometry Author Reinhold Baer
ISBN-10 9780486154664
Release 2012-06-11
Pages 336
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Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. 1952 edition.

Mathematical Models

Mathematical Models Author Gerd Fischer
ISBN-10 9783658188658
Release 2017-09-04
Pages 216
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This book presents beautiful photos of mathematical models of geometric surfaces made from a variety of materials including plaster, metal, paper, wood, and string. The construction of these models at the time (of Felix Klein and others) was not an end in itself, but was accompanied by mathematical research especially in the field of algebraic geometry. The models were used to illustrate the mathematical objects defined by abstract formulas, either as equations or parameterizations. In the second part of the book, the models are explained by experts in the field of geometry. This book is a reprint thirty years after the original publication in 1986 with a new preface by Gert-Martin Greuel. The models have a timeless appeal and a historical value.

An outline of projective geometry

An outline of projective geometry Author Lynn E. Garner
ISBN-10 UOM:39015015621587
Release 1981
Pages 220
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An outline of projective geometry has been writing in one form or another for most of life. You can find so many inspiration from An outline of projective geometry also informative, and entertaining. Click DOWNLOAD or Read Online button to get full An outline of projective geometry book for free.

An Introduction to Finite Projective Planes

An Introduction to Finite Projective Planes Author Abraham Adrian Albert
ISBN-10 9780486789941
Release 2015-02-18
Pages 112
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Text for both beginning and advanced undergraduate and graduate students covers finite planes, field planes, coordinates in an arbitrary plane, central collineations and the little Desargues' property, the fundamental theorem, and non-Desarguesian planes. 1968 edition.

A Guide to the Classification Theorem for Compact Surfaces

A Guide to the Classification Theorem for Compact Surfaces Author Jean Gallier
ISBN-10 9783642343643
Release 2013-02-05
Pages 178
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This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example. Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.

Homotopy Theory of the Suspensions of the Projective Plane

Homotopy Theory of the Suspensions of the Projective Plane Author Jie Wu
ISBN-10 9780821832394
Release 2003
Pages 130
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The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds.

Oriented Projective Geometry

Oriented Projective Geometry Author Jorge Stolfi
ISBN-10 9781483265193
Release 2014-05-10
Pages 246
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Oriented Projective Geometry: A Framework for Geometric Computations proposes that oriented projective geometry is a better framework for geometric computations than classical projective geometry. The aim of the book is to stress the value of oriented projective geometry for practical computing and develop it as a rich, consistent, and effective tool for computer programmers. The monograph is comprised of 20 chapters. Chapter 1 gives a quick overview of classical and oriented projective geometry on the plane, and discusses their advantages and disadvantages as computational models. Chapters 2 through 7 define the canonical oriented projective spaces of arbitrary dimension, the operations of join and meet, and the concept of relative orientation. Chapter 8 defines projective maps, the space transformations that preserve incidence and orientation; these maps are used in chapter 9 to define abstract oriented projective spaces. Chapter 10 introduces the notion of projective duality. Chapters 11, 12, and 13 deal with projective functions, projective frames, relative coordinates, and cross-ratio. Chapter 14 tells about convexity in oriented projective spaces. Chapters 15, 16, and 17 show how the affine, Euclidean, and linear vector spaces can be emulated with the oriented projective space. Finally, chapters 18 through 20 discuss the computer representation and manipulation of lines, planes, and other subspaces. Computer scientists and programmers will find this text invaluable.

Projective Geometry

Projective Geometry Author Source Wikipedia
ISBN-10 1157690572
Release 2010-06
Pages 370
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 119. Chapters: Projective plane, Stereographic projection, Hyperplane, M bius transformation, Projective linear group, Homogeneous coordinates, Projective space, Pl cker coordinates, Complex projective space, Riemann sphere, Cross-ratio, Fubini-Study metric, SL2(R), Duality, Grassmannian, Real projective line, Projective orthogonal group, Five points determine a conic, Inverse curve, 3D projection, Dual curve, Direct linear transformation, Desargues' theorem, Cayley-Bacharach theorem, Real projective space, Pascal's theorem, Fano plane, Inversive ring geometry, Semilinear transformation, Bloch sphere, PSL(2,7), Collineation, Pole and polar, Incidence, Homography, Pappus's hexagon theorem, Quadric, Near-field, Line at infinity, Projective harmonic conjugate, Schwarzian derivative, Differential invariant, Segre embedding, Oval, Complete quadrangle, Gnomonic projection, Pentagram map, Plane at infinity, Quaternionic projective space, Translation plane, Planar ternary ring, Affine Grassmannian, Cayley-Klein metric, Oriented projective geometry, Complex projective plane, Point at infinity, Hyperplane at infinity, Intersection theorem, Maximal arc, Projective frame, Imaginary line, Projectivization, Brianchon's theorem, Braikenridge-Maclaurin theorem, Moufang plane, W-curve, Desmic system, Klein quadric, Projective differential geometry, Birkhoff-Grothendieck theorem, Real point, Circular points at infinity, Projective cone, Non-Desarguesian plane, Cayley plane, Ovoid, Isotropic line, Hughes plane, Correlation, Polar hypersurface, Imaginary point, Reciprocity, Imaginary curve, Real curve. Excerpt: In geometry, a M bius transformation of the plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad ? bc ? 0. M bius transformations are named in honor of August F...

Introduction to Projective Geometry

Introduction to Projective Geometry Author C. R. Wylie
ISBN-10 9780486141701
Release 2011-09-12
Pages 576
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This introductory volume offers strong reinforcement for its teachings, with detailed examples and numerous theorems, proofs, and exercises, plus complete answers to all odd-numbered end-of-chapter problems. 1970 edition.

Multiple View Geometry in Computer Vision

Multiple View Geometry in Computer Vision Author Richard Hartley
ISBN-10 9781139449144
Release 2004-03-25
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A basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction covering the key ideas in the book (which itself has been updated with additional examples and appendices) and significant new results which have appeared since the first edition. Comprehensive background material is provided, so readers familiar with linear algebra and basic numerical methods can understand the projective geometry and estimation algorithms presented, and implement the algorithms directly from the book.

A Modern View of Geometry

A Modern View of Geometry Author Leonard M. Blumenthal
ISBN-10 9780486821139
Release 2017-04-19
Pages 208
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Elegant exposition of postulation geometry of planes offers rigorous, lucid treatment of coordination of affine and projective planes, set theory, propositional calculus, affine planes with Desargues and Pappus properties, more. 1961 edition.

Foundations of Projective Geometry

Foundations of Projective Geometry Author Robin Hartshorne
ISBN-10 4871878376
Release 2009-12-01
Pages 190
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The first geometrical properties of a projective nature were discovered in the third century by Pappus of Alexandria. Filippo Brunelleschi (1404-1472) started investigating the geometry of perspective in 1425. Johannes Kepler (1571-1630) and Gerard Desargues (1591-1661) independently developed the pivotal concept of the "point at infinity." Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-years old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy in 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.

Projective Geometry

Projective Geometry Author Albrecht Beutelspacher
ISBN-10 0521483646
Release 1998-01-29
Pages 258
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A textbook on projective geometry that emphasises applications in modern information and communication science.