# Elementary Geometry 3 Congruence axioms 3.1 The length of a line segment Affine geometry by itself does not capture the full richness of our intuitive picture ... Affine geometry enriched by means of this additional structure is called Euclidean geometry .

Author: John Roe

Publisher: Clarendon Press

ISBN: 0198534566

Category: Mathematics

Page: 307

View: 327

This text is a careful introduction to geometry. While developing geometry, the book also emphasizes the links between geometry and other branches of pure and applied mathematics.
Categories: Mathematics

# Metric Affine Geometry Metric Affine Geometry focuses on linear algebra, which is the source for the axiom systems of all affine and projective geometries, both metric and nonmetric. This book is organized into three chapters.

Author: Ernst Snapper

Publisher: Elsevier

ISBN: 9781483269337

Category: Mathematics

Page: 456

View: 899

Metric Affine Geometry focuses on linear algebra, which is the source for the axiom systems of all affine and projective geometries, both metric and nonmetric. This book is organized into three chapters. Chapter 1 discusses nonmetric affine geometry, while Chapter 2 reviews inner products of vector spaces. The metric affine geometry is treated in Chapter 3. This text specifically discusses the concrete model for affine space, dilations in terms of coordinates, parallelograms, and theorem of Desargues. The inner products in terms of coordinates and similarities of affine spaces are also elaborated. The prerequisites for this publication are a course in linear algebra and an elementary course in modern algebra that includes the concepts of group, normal subgroup, and quotient group. This monograph is suitable for students and aspiring geometry high school teachers.
Categories: Mathematics

# The Reduction of Physical Theories Many Euclidean spaces have the same affine space in the sense of (4.77), and every affine space is the affine space of a certain Euclidean space in the sense of (4.78). In this sense, Euclidean geometry is a refinement of affine ...

Author: Erhard Scheibe

Publisher: Springer Nature

ISBN: 9783662650004

Category: Science

Page: 235

View: 679

Using simple physical examples, this work by Erhard Scheibe presents an important and powerful approach to the reduction of physical theories. Novel to the approach is that it is not based, as usual, on a single reduction concept that is fixed once and for all, but on a series of recursively constructed reductions, with which all reductions appear as combinations of very specific elementary reductions. This leaves the general notion of theory reduction initially open and is beneficial for the treatment of the difficult cases of reduction from the fields of special and general relativity, thermodynamics, statistical mechanics,and quantum mechanics, which are treated in the second volume. The book is systematically organized and intended for readers interested in philosophy of science as well as physicists without deep philosophical knowledge.
Categories: Science

# Computing in Euclidean Geometry Thus Euclidean geometry ( i.e. , R ? ) is an affine geometry ; i.e. , Euclidean geometry is a model of the theory of affine geometry . Of course , this model has much richer structures than an ordinary affine geometry .

Author: Dingzhu Du

Publisher: World Scientific

ISBN: 9810209665

Category: Mathematics

Page: 385

View: 843

This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. The topics covered are: a history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra; triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and steiner trees. Each chapter is written by a leading expert in the field and together they provide a clear and authoritative picture of what computational Euclidean geometry is and the direction in which research is going.
Categories: Mathematics

# Differential Geometry The method of interpretation of these phrases employed here is modeled after that of Cartan ( C.2 ] , ' in which the analogy with affine and Euclidean geometry is developed by treating these geometries themselves in an invariant way in ...

Author: J. J. Stoker

Publisher: John Wiley & Sons

ISBN: 0471504033

Category: Mathematics

Page: 432

View: 195

This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.
Categories: Mathematics

# Computing in Euclidean Geometry Thus Euclidean geometry ( i.e. , R ? ) is an affine geometry ; i.e. , Euclidean geometry is a model of the theory of affine geometry . Of course , this model has much richer structures than an ordinary affine geometry .

Author: Ding-Zhu Du

Publisher: World Scientific

ISBN: 9810218761

Category: Mathematics

Page: 492

View: 550

This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. Topics covered include the history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra, triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and Steiner trees. This second edition contains three new surveys covering geometric constraint solving, computational geometry and the exact computation paradigm.
Categories: Mathematics

# Euclidean and Non Euclidean Geometry International Student Edition The subset of affine geometry consisting of those facts of Euclidean geometry that continue to make sense when the figure in question is subjected to transformations by the Galilean group is called Galilean geometry and is the subject ...

Author: Patrick J. Ryan

Publisher: Cambridge University Press

ISBN: 9780521127073

Category: Mathematics

Page: 232

View: 326

This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic.
Categories: Mathematics