This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24.

Author: I. M. Yaglom

Publisher: MAA

ISBN: 0883856484

Category: Mathematics

Page: 285

View: 704

The familiar plane geometry of secondary school - figures composed of lines and circles - takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincar model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in elementary geometry and trigonometry.

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane.

Author: Vladimir V Kisil

Publisher: World Scientific

ISBN: 9781908977601

Category: Mathematics

Page: 208

View: 614

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL2(R). Starting from elementary facts in group theory, the author unveils surprising new results about the geometry of circles, parabolas and hyperbolas, using an approach based on the Erlangen programme of F Klein, who defined geometry as a study of invariants under a transitive group action.The treatment of elliptic, parabolic and hyperbolic Möbius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers which represent all non-isomorphic commutative associative two-dimensional algebras with unit. The hypercomplex numbers are in perfect correspondence with the three types of geometries concerned. Furthermore, connections with the physics of Minkowski and Galilean space-time are considered./a

Your study of geometry has just begun. Now you are starting to do geometry, and
your geometric education should be continued, both formally and informally. And
it should be reasonably continuous. An occasional course in some aspect of ...

Author: Clayton W. Dodge

Publisher: Courier Corporation

ISBN: 0486434761

Category: Mathematics

Page: 295

View: 359

This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane.

Author: Vladimir V. Kisil

Publisher: World Scientific

ISBN: 9781848168589

Category: Mathematics

Page: 192

View: 539

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL2(R). Starting from elementary facts in group theory, the author unveils surprising new results about the geometry of circles, parabolas and hyperbolas, using an approach based on the Erlangen programme of F Klein, who defined geometry as a study of invariants under a transitive group action.The treatment of elliptic, parabolic and hyperbolic Möbius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers which represent all non-isomorphic commutative associative two-dimensional algebras with unit. The hypercomplex numbers are in perfect correspondence with the three types of geometries concerned. Furthermore, connections with the physics of Minkowski and Galilean space-time are considered.

Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software ...

Author: Judith N. Cederberg

Publisher: Springer Science & Business Media

ISBN: 9781475734904

Category: Mathematics

Page: 441

View: 221

Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclids and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometers Sketchpad".

This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III.

Author: I. M. Yaglom

Publisher: Mathematical Association of America

ISBN: 0883856484

Category: Mathematics

Page: 293

View: 940

The familiar plane geometry of secondary school - figures composed of lines and circles - takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincaré model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in elementary geometry and trigonometry.

Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.

Author: Bruce E. Meserve

Publisher: Courier Corporation

ISBN: 9780486152264

Category: Mathematics

Page: 336

View: 805

Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.

Author: Ana Irene Ramírez GalarzaPublish On: 2007-05-02

This book develops the geometric intuition of the reader by examining the symmetries (or rigid motions) of the space in question.

Author: Ana Irene Ramírez Galarza

Publisher: Springer Science & Business Media

ISBN: 3764375183

Category: Mathematics

Page: 220

View: 666

This book develops the geometric intuition of the reader by examining the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. The main focus is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. Basic notions of algebra and analysis are used to convey better understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.

In this edition, Smart covers the major new applied areas of computer graphics, and emphasizes matrices for transformations.

Author: James R. Smart

Publisher: Brooks Cole

ISBN: PSU:000021990076

Category: Mathematics

Page: 410

View: 138

This comprehensive, best-selling text focuses on the study of many different geometries--rather than a single geometry--and emphasizes practical applications. Designed to be a flexible teaching tool for a wide range of students (including math, education, or computer science majors), Smart's text features self-contained chapters organized so that instructors can cover as much or as little of each topic as they choose, from bare minimum one-section coverage to full-chapter coverage. Modern Geometries has earned a reputation for its logical progression of ideas, its well-constructed exercises, and its comprehensive coverage. In this edition, Smart covers the major new applied areas of computer graphics, and emphasizes matrices for transformations.

The Geometry of Solitons : AARMS-CRM Workshop, June 4-9, 1999, Halifax, N.S.
, Canada A. A. Coley. CRM Proceedings ... Alternatively, one may generate
discrete geometries by means of Backlund transformations. However, a priori, it is
by ...

Author: A. A. Coley

Publisher: American Mathematical Soc.

ISBN: 0821870254

Category: Mathematics

Page: 436

View: 561

This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing methodand Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductorylectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.

This book, nearly a century after its initial publication, remains a very approachable and understandable treatment of the subject.

Author: John Wesley Young

Publisher: American Mathematical Soc.

ISBN: 9781614440048

Category: Geometry, Projective

Page: 185

View: 517

John Wesley Young co-authored with Oswald Veblen the first monograph on projective geometry in English. That careful and thorough axiomatic treatment remains read today. This volume is Young's attempt to write an accessible and intuitive treatment for non-specialists. The first five chapters are a careful and elementary treatment of the subject culminating in the theorems of Pascal and Brianchon and the polar system of a conic. Later chapters pull metric consequences from projective results and consider the Kleinian classification of geometries by their groups of transformations. This book, nearly a century after its initial publication, remains a very approachable and understandable treatment of the subject.

Geometry of Real Inner Product Spaces Walter Benz ... The close connection
between Lorentz transformations (see section 17 of chapter 3) and Lie transformations (section 12), more precisely Laguerre transformations (section 13
), has been ...

Author: Walter Benz

Publisher: Springer Science & Business Media

ISBN: 3764374322

Category: Mathematics

Page: 244

View: 905

Presents the real inner product spaces of arbitrary (finite or infinite) dimension greater than or equal to 2. This book studies the sphere geometries of Mobius and Lie for these spaces, besides euclidean and hyperbolic geometry, as well as geometries where Lorentz transformations play the key role.

B|BLIOGRAPHY See the Bibliography of Chapter 1; also: BARRY, E. H.,
Introduction to Geometrical Transformations. Boston, Mass.: Prindle ... Coxford,
A. F., and Z. P. Usiskin, Geometry, a Transformation Approach. River Forest, Ill.:
Laidlaw ...

Author: Howard Whitley Eves

Publisher: Jones & Bartlett Learning

ISBN: 0867204753

Category: Mathematics

Page: 370

View: 943

College Geometry is divided into two parts. Part I is a sequel to basic high school geometry and introduces the reader to some of the important modern extensions of elementary geometry- extension that have largely entered into the mainstream of mathematics. Part II treats notions of geometric structure that arose with the non-Euclidean revolution in the first half of the nineteenth century.

In the present case however, we are given a certain group of transformations, the
modular group, and we would like to construct a geometry, that is, define a
distance, in such a way that transformations of this group are motions. In this
situation ...

Author: Viacheslav V. Nikulin

Publisher: Springer Science & Business Media

ISBN: 3540152814

Category: Mathematics

Page: 251

View: 404

This is a quite exceptional book, a lively and approachable treatment of an important field of mathematics given in a masterly style. Assuming only a school background, the authors develop locally Euclidean geometries, going as far as the modular space of structures on the torus, treated in terms of Lobachevsky's non-Euclidean geometry. Each section is carefully motivated by discussion of the physical and general scientific implications of the mathematical argument, and its place in the history of mathematics and philosophy. The book is expected to find a place alongside classics such as Hilbert and Cohn-Vossen's "Geometry and the imagination" and Weyl's "Symmetry".

Undergraduate-level introduction to linear algebra and matrix theory.

Author: Charles G. Cullen

Publisher: Courier Corporation

ISBN: 0486663280

Category: Mathematics

Page: 318

View: 736

Undergraduate-level introduction to linear algebra and matrix theory. Explores matrices and linear systems, vector spaces, determinants, spectral decomposition, Jordan canonical form, much more. Over 375 problems. Selected answers. 1972 edition.