Collineations and Conic Sections
Collineations and Conic Sections

Author: Christopher Baltus

Publisher: Springer Nature

Published: 2020-09-01

Total Pages: 190

ISBN-13: 3030462870

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This volume combines an introduction to central collineations with an introduction to projective geometry, set in its historical context and aiming to provide the reader with a general history through the middle of the nineteenth century. Topics covered include but are not limited to: The Projective Plane and Central Collineations The Geometry of Euclid's Elements Conic Sections in Early Modern Europe Applications of Conics in History With rare exception, the only prior knowledge required is a background in high school geometry. As a proof-based treatment, this monograph will be of interest to those who enjoy logical thinking, and could also be used in a geometry course that emphasizes projective geometry.

Projective Geometry
Projective Geometry

Author: Elisabetta Fortuna

Publisher: Springer

Published: 2016-12-17

Total Pages: 275

ISBN-13: 3319428241

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This book starts with a concise but rigorous overview of the basic notions of projective geometry, using straightforward and modern language. The goal is not only to establish the notation and terminology used, but also to offer the reader a quick survey of the subject matter. In the second part, the book presents more than 200 solved problems, for many of which several alternative solutions are provided. The level of difficulty of the exercises varies considerably: they range from computations to harder problems of a more theoretical nature, up to some actual complements of the theory. The structure of the text allows the reader to use the solutions of the exercises both to master the basic notions and techniques and to further their knowledge of the subject, thus learning some classical results not covered in the first part of the book. The book addresses the needs of undergraduate and graduate students in the theoretical and applied sciences, and will especially benefit those readers with a solid grasp of elementary Linear Algebra.

Projective Geometry
Projective Geometry

Author: H.S.M. Coxeter

Publisher: Springer Science & Business Media

Published: 2003-10-09

Total Pages: 180

ISBN-13: 9780387406237

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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.

Projective and Cayley-Klein Geometries
Projective and Cayley-Klein Geometries

Author: Arkadij L. Onishchik

Publisher: Springer Science & Business Media

Published: 2006-11-22

Total Pages: 445

ISBN-13: 3540356452

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This book offers an introduction into projective geometry. The first part presents n-dimensional projective geometry over an arbitrary skew field; the real, the complex, and the quaternionic geometries are the central topics, finite geometries playing only a minor part. The second deals with classical linear and projective groups and the associated geometries. The final section summarizes selected results and problems from the geometry of transformation groups.

Projective Geometry
Projective Geometry

Author: Pierre Samuel

Publisher: Springer

Published: 1988-09-12

Total Pages: 180

ISBN-13:

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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

Projective Duality and Homogeneous Spaces
Projective Duality and Homogeneous Spaces

Author: Evgueni A. Tevelev

Publisher: Springer Science & Business Media

Published: 2006-03-30

Total Pages: 257

ISBN-13: 3540269576

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Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. Thus the appearance of a book specifically devoted to projective duality is a long-awaited and welcome event. Projective Duality and Homogeneous Spaces covers a vast and diverse range of topics in the field of dual varieties, ranging from differential geometry to Mori theory and from topology to the theory of algebras. It gives a very readable and thorough account and the presentation of the material is clear and convincing. For the most part of the book the only prerequisites are basic algebra and algebraic geometry. This book will be of great interest to graduate and postgraduate students as well as professional mathematicians working in algebra, geometry and analysis.

Basic Algebraic Geometry 2
Basic Algebraic Geometry 2

Author: Igor Rostislavovich Shafarevich

Publisher: Springer Science & Business Media

Published: 1994

Total Pages: 292

ISBN-13: 9783540575542

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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.

Perspectives on Projective Geometry
Perspectives on Projective Geometry

Author: Jürgen Richter-Gebert

Publisher: Springer Science & Business Media

Published: 2011-02-04

Total Pages: 573

ISBN-13: 3642172865

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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.

An Undergraduate Primer in Algebraic Geometry
An Undergraduate Primer in Algebraic Geometry

Author: Ciro Ciliberto

Publisher: Springer Nature

Published: 2021-05-05

Total Pages: 327

ISBN-13: 3030710211

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This book consists of two parts. The first is devoted to an introduction to basic concepts in algebraic geometry: affine and projective varieties, some of their main attributes and examples. The second part is devoted to the theory of curves: local properties, affine and projective plane curves, resolution of singularities, linear equivalence of divisors and linear series, Riemann–Roch and Riemann–Hurwitz Theorems. The approach in this book is purely algebraic. The main tool is commutative algebra, from which the needed results are recalled, in most cases with proofs. The prerequisites consist of the knowledge of basics in affine and projective geometry, basic algebraic concepts regarding rings, modules, fields, linear algebra, basic notions in the theory of categories, and some elementary point–set topology. This book can be used as a textbook for an undergraduate course in algebraic geometry. The users of the book are not necessarily intended to become algebraic geometers but may be interested students or researchers who want to have a first smattering in the topic. The book contains several exercises, in which there are more examples and parts of the theory that are not fully developed in the text. Of some exercises, there are solutions at the end of each chapter.