Perspectives on Projective Geometry

Author: Jürgen Richter-Gebert
Publisher: Springer Science & Business Media
ISBN: 9783642172861
Format: PDF
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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.

Perspectives on Projective Geometry

Author: Jürgen Richter-Gebert
Publisher: Springer
ISBN: 9783642172854
Format: PDF
Download and Read
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.

Affine and Projective Geometry

Author: M. K. Bennett
Publisher: John Wiley & Sons
ISBN: 1118030826
Format: PDF, ePub, Docs
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An important new perspective on AFFINE AND PROJECTIVE GEOMETRY This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view. Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises, covering material for two semesters of upper-level undergraduate mathematics. The first part of the book deals with the correlation between synthetic geometry and linear algebra. In the second part, geometry is used to introduce lattice theory, and the book culminates with the fundamental theorem of projective geometry. While emphasizing affine geometry and its basis in Euclidean concepts, the book: * Builds an appreciation of the geometric nature of linear algebra * Expands students' understanding of abstract algebra with its nontraditional, geometry-driven approach * Demonstrates how one branch of mathematics can be used to prove theorems in another * Provides opportunities for further investigation of mathematics by various means, including historical references at the ends of chapters Throughout, the text explores geometry's correlation to algebra in ways that are meant to foster inquiry and develop mathematical insights whether or not one has a background in algebra. The insight offered is particularly important for prospective secondary teachers who must major in the subject they teach to fulfill the licensing requirements of many states. Affine and Projective Geometry's broad scope and its communicative tone make it an ideal choice for all students and professionals who would like to further their understanding of things mathematical.

Introduction to Projective Geometry

Author: C. R. Wylie
Publisher: Courier Corporation
ISBN: 0486141705
Format: PDF
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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.

Projective Geometry

Author: Rey Casse
Publisher: Oxford University Press, USA
ISBN:
Format: PDF, ePub, Docs
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This lucid and accessible text provides an introductory guide to projective geometry, an area of mathematics concerned with the properties and invariants of geometric figures under projection. Including numerous worked examples and exercises throughout, the book covers axiomatic geometry, field planes and PG(r, F), coordinating a projective plane, non-Desarguesian planes, conics and quadrics in PG(3, F). Assuming familiarity with linear algebra, elementary group theory, partial differentiation and finite fields, as well as some elementary coordinate geometry, this text is ideal for 3rd and 4th year mathematics undergraduates.

Projective Geometry

Author: H.S.M. Coxeter
Publisher: Springer Science & Business Media
ISBN: 9780387406237
Format: PDF, Docs
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In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.

Symmetry and Pattern in Projective Geometry

Author: Eric Lord
Publisher: Springer Science & Business Media
ISBN: 144714631X
Format: PDF, ePub, Mobi
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Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates, and brief introductions to Plücker coordinates and Grassmann coordinates are presented. This book looks carefully at linear, quadratic, cubic and quartic figures in two, three and higher dimensions. It deals at length with the extensions and consequences of basic theorems such as those of Pappus and Desargues. The emphasis throughout is on special configurations that have particularly interesting symmetry properties. The intricate and novel ideas of ‘Donald’ Coxeter, who is considered one of the great geometers of the twentieth century, are also discussed throughout the text. The book concludes with a useful analysis of finite geometries and a description of some of the remarkable configurations discovered by Coxeter. This book will be appreciated by mathematics students and those wishing to learn more about the subject of geometry. It makes accessible subjects and theorems which are often considered quite complicated and presents them in an easy-to-read and enjoyable manner.

Visual Perception Theory and Practice

Author: Terry Caelli
Publisher: Elsevier
ISBN: 1483189147
Format: PDF
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Visual Perception: Theory and Practice focuses on the theory and practice of visual perception, with emphasis on technologies used in vision research and in visual information processing. Central areas of vision research including spatial vision, motion perception, and color are discussed. Light and optics, convolutions and Fourier methods, and network theory and systems are also examined. Comprised of nine chapters, this book begins with an overview of language and processes underlying specific areas of vision such as measures of neural activity, feature specificity, and individual cells and psychophysics. The reader is then systematically introduced to the more essential properties of light and optics relevant to visual perception; the use of convolutions, Fourier series, and Fourier transform to model processes in visual perception; and network theory and systems. Subsequent chapters deal with the geometry of visual perception; spatial vision; the perception of motion; and some specific issues in visual perception, including color perception, binocular vision, and steriopsis. This monograph is intended for students, practitioners, and investigators in physiology.

Viewpoints

Author: Marc Frantz
Publisher: Princeton University Press
ISBN: 9781400839056
Format: PDF, Mobi
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An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students. Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery. Classroom-tested activities and problem solving Accessible problems that move beyond regular art school curriculum Multiple solutions of varying difficulty and applicability Appropriate for students of all mathematics and art levels Original and exclusive essays by contemporary artists Forthcoming: Instructor's manual (available only to teachers)

Shadows of Reality

Author: Tony Robbin
Publisher: Yale University Press
ISBN: 9780300129625
Format: PDF, ePub
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In this insightful book, which is a revisionist math history as well as a revisionist art history, Tony Robbin, well known for his innovative computer visualizations of hyperspace, investigates different models of the fourth dimension and how these are applied in art and physics. Robbin explores the distinction between the slicing, or Flatland, model and the projection, or shadow, model. He compares the history of these two models and their uses and misuses in popular discussions. Robbin breaks new ground with his original argument that Picasso used the projection model to invent cubism, and that Minkowski had four-dimensional projective geometry in mind when he structured special relativity. The discussion is brought to the present with an exposition of the projection model in the most creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. Robbin clarifies these esoteric concepts with understandable drawings and diagrams. Robbin proposes that the powerful role of projective geometry in the development of current mathematical ideas has been long overlooked and that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime. He offers a fascinating review of how projective ideas are the source of some of today’s most exciting developments in art, math, physics, and computer visualization.