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Author: Jürgen Richter-Gebert Publisher: Springer Science & Business Media ISBN: 3642172865 Category : Mathematics Languages : en Pages : 573
Book Description
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.
Author: Jürgen Richter-Gebert Publisher: Springer Science & Business Media ISBN: 3642172865 Category : Mathematics Languages : en Pages : 573
Book Description
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.
Author: H.S.M. Coxeter Publisher: Springer Science & Business Media ISBN: 9780387406237 Category : Mathematics Languages : en Pages : 180
Book Description
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.
Author: C. R. Wylie Publisher: Courier Corporation ISBN: 0486141705 Category : Mathematics Languages : en Pages : 578
Book Description
This lucid introductory text offers both an analytic and an axiomatic approach to plane projective geometry. The analytic treatment builds and expands upon students' familiarity with elementary plane analytic geometry and provides a well-motivated approach to projective geometry. Subsequent chapters explore Euclidean and non-Euclidean geometry as specializations of the projective plane, revealing the existence of an infinite number of geometries, each Euclidean in nature but characterized by a different set of distance- and angle-measurement formulas. Outstanding pedagogical features include worked-through examples, introductions and summaries for each topic, and numerous theorems, proofs, and exercises that reinforce each chapter's precepts. Two helpful indexes conclude the text, along with answers to all odd-numbered exercises. In addition to its value to undergraduate students of mathematics, computer science, and secondary mathematics education, this volume provides an excellent reference for computer science professionals.
Author: T. Ewan Faulkner Publisher: Courier Corporation ISBN: 0486154890 Category : Mathematics Languages : en Pages : 148
Book Description
Highlighted by numerous examples, this book explores methods of the projective geometry of the plane. Examines the conic, the general equation of the 2nd degree, and the relationship between Euclidean and projective geometry. 1960 edition.
Author: Oswald Veblen Publisher: Franklin Classics ISBN: 9780342703401 Category : Languages : en Pages : 362
Book Description
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Author: Reinhold Baer Publisher: Courier Corporation ISBN: 0486154661 Category : Mathematics Languages : en Pages : 338
Book Description
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.
Author: Oswald Veblen Publisher: Rarebooksclub.com ISBN: 9781230097206 Category : Languages : en Pages : 110
Book Description
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 edition. Excerpt: ...The coordinates of a double point (xv x) must satisfy the equations pxi = ax1 + bx2, px = cx1 + dxr These equations are compatible only if the determinant of the system 3 (a-p)c1+bx2=0, cxl+(d-p)x =0, vanishes. This leads to the equation a-p b I' c d--p for the determination of the factor of proportionality p. This equation is called the characteristic equation of the matrix representing the projectivity. Every value of p satisfying this equation then leads to a double point when substituted in one of the equations (3); viz., if p, be a solution of the characteristic equation, the point (xv ag = (-b, a-pl) = (d-pv-c) is a double point. In homogeneous coordinates the cross ratio R (AB, CD) of four points A = (av a3), B = (bv ft2), C = (cv c2), D = (dv d ) is given by (ac) (be) (ad) ' (bd)' where the expressions (ac), etc., are used as abbreviations for a--a, etc. This statement is readily verified by writing down the above ratio in terms of the nonhomogeneous coordinates of the four points. We will close this section by giving to the two homogeneous coordinates of a point on a line an explicit geometrical significance. In view of the fact that the coordinates of such a point are not uniquely determined, a factor of proportionality being entirely arbitrary, there may be many such interpretations. On account of the existence of this arbitrary factor, we may impose a further condition on the coordinates (xv x2) of a point, in addition to the defining relation x1/x2=x, where x is the nonhomogeneous coordinate of the point in question. We choose the relation x1 + x2 = 1. If this relation is satisfied, Thus homogeneous coordinates subject to the condition xl+x2 = l can be defined by choosing three points A, B, C arbitrarily, ...
Author: Igor Rostislavovich Shafarevich Publisher: Springer Science & Business Media ISBN: 9783540575542 Category : Mathematics Languages : en Pages : 292
Book Description
The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with Volume 1 the author has revised the text and added new material, e.g. a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as in theoretical physics.
Author: Pierre Samuel Publisher: Springer ISBN: Category : Mathematics Languages : en Pages : 180
Book Description
The purpose of this book is to revive some of the beautiful results obtained by various geometers of the 19th century, and to give its readers a taste of concrete algebraic geometry. A good deal of space is devoted to cross-ratios, conics, quadrics, and various interesting curves and surfaces. The fundamentals of projective geometry are efficiently dealt with by using a modest amount of linear algebra. An axiomatic characterization of projective planes is also given. While the topology of projective spaces over real and complex fields is described, and while the geometry of the complex projective libe is applied to the study of circles and Möbius transformations, the book is not restricted to these fields. Interesting properties of projective spaces, conics, and quadrics over finite fields are also given. This book is the first volume in the Readings in Mathematics sub-series of the UTM. From the reviews: "...The book of P. Samuel thus fills a gap in the literature. It is a little jewel. Starting from a minimal background in algebra, he succeeds in 160 pages in giving a coherent exposition of all of projective geometry. ... one reads this book like a novel. " D.Lazard in Gazette des Mathématiciens#1
Author: Jürgen Richter-Gebert
Author: Jürgen Richter-Gebert
Author: Oswald Veblen
Author: H.S.M. Coxeter
Author: C. R. Wylie
Author: T. Ewan Faulkner
Author: Oswald Veblen
Author: Reinhold Baer
Author: Oswald Veblen
Author: Igor Rostislavovich Shafarevich
Author: Pierre Samuel