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2007 | Buch

Introduction to Classical Geometries

verfasst von: Ana Irene Ramírez Galarza, José Seade

Verlag: Birkhäuser Basel

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Über dieses Buch

Geometry is one of the oldest branches of mathematics, nearly as old as human culture.Itsbeautyhasalwaysfascinatedmathematicians,amongothers.Inwriting this book we had the purpose of sharing with readers the pleasure derived from studying geometry,as well as giving a tasteof its importance, its deep connections with other branches of mathematics and the highly diverse viewpoints that may be taken by someone entering this ?eld. We also want to propose a speci?c way to introduce concepts that have arisen from the heyday of the Greek school of geometry to the present day. We workwithcoordinatemodels,sincethis facilitatestheuseofalgebraicandanalytic results, and we follow the viewpoint proposed by Felix Klein in the 19th century, of studying geometry via groups of symmetries of the space in question. We intend this book to be both an introduction to the subject addressed to undergraduate students in mathematics and physics, and a useful text-book for mathematicians and scientists in general who want to learn the basics of classical geometry: Euclidean, a?ne, elliptic, hyperbolic and projective geometry. These are all presented in a uni?ed way and the essential content of this book may be covered in a single semester, though a longer period of study would allow the student to grasp and assimilate better the material in it.

Inhaltsverzeichnis

Frontmatter
1. Euclidean geometry
Abstract
The branch of mathematics known as geometry was formally established in Greece around the year 300 B.C. However its origins, for our Western culture, go back to Mesopotamia and Egypt around the year 3000 B.C.
2. Affine geometry
Abstract
This exposition of affine geometry is somehow different from those usually found in the literature. We have chosen a way of presenting affine geometry that constitutes a natural bridge between Euclidean geometry and projective geometry, both from the historic and the formal viewpoints. The reason this is possible is that the group of affine transformations is larger than the Euclidean group and is contained in the group of projective transformations.
3. Projective geometry
Abstract
Projective geometry is the most extensive geometry among those we study in this book: its group of transformations admits as subgroups those we have already studied and those we shall study later on.
4. Hyperbolic geometry
Abstract
In Chapter 3 we saw that elliptic geometry is a non-Euclidean geometry, for any pair of elliptic lines intersect; that is, parallel lines do not exist in that geometry (denial N1 of Postulate V).
5. Appendices
Abstract
To extend the concept of differentiability to a function of several variables, F : ℝn→ℝ, at a point P0 of its domain, we must recall that the value of a differentiable function at a point close to P0, P0 + \( \bar h \), can be approximated by the value of the function at the point, F(P0), plus the value of a linear transformation, called precisely the derivative at P0, applied to the increment \( \bar h \), where \( \left\| {\bar h} \right\| \) is sufficiently small.
Backmatter
Metadaten
Titel
Introduction to Classical Geometries
verfasst von
Ana Irene Ramírez Galarza
José Seade
Copyright-Jahr
2007
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-7643-7518-8
Print ISBN
978-3-7643-7517-1
DOI
https://doi.org/10.1007/978-3-7643-7518-8