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## Über dieses Buch

Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michèle Audin, professor at the University of Strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces.
It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for most of the exercises are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding.

## Inhaltsverzeichnis

### Introduction

Abstract
This is a book written for students who have been taught a small amount of geometry at secondary school and some linear algebra at university. It comes from several courses I have taught in Strasbourg.
Michèle Audin

### Chapter I. Affine Geometry

Abstract
An affine space is a set of points; it contains lines, etc. and affine geometry(1) deals, for instance, with the relations between these points and these lines (collinear points, parallel or concurrent lines…). To define these objects and describe their relations, one can:
• Either state a list of axioms, describing incidence properties, like “through two points passes a unique line”. This is the way followed by Euclid (and more recently by Hilbert). Even if the process and a fortiori the axioms themselves are not explicitly stated, this is the way used in secondary schools.
• Or decide that the essential thing is that two points define a vector and define everything starting from linear algebra, namely from the axioms defining the vector spaces.
Michèle Audin

### Chapter II. Euclidean Geometry, Generalities

Abstract
In this chapter, all the vector spaces are defined over the field R of real numbers. The spaces under consideration all have finite dimension.
Michèle Audin

### Chapter III. Euclidean Geometry in the Plane

Abstract
In this chapter, there are plane isometries, triangles and angles at the circumference, similarities, inversions and even pencils of circles. But there is also, and we are forced to begin with this, a discussion of what an angle is and how to measure it. The proofs are of course very simple but the statements and their precision are subtle and important.
Michèle Audin

### Chapter IV. Euclidean Geometry in Space

Abstract
In this chapter, everything will take place in a Euclidean (affine or vector) space of dimension 3.
Michèle Audin

### Chapter V. Projective Geometry

Abstract
Projective geometry was created as a completion of affine geometry in which there are no parallels, which is a nice property, as it gives neater statements.
Michèle Audin

### Chapter VI. Conics and Quadrics

Abstract
This chapter is devoted to quadrics and especially to conics. I have tried to keep a balance between:
• the algebraic aspects: a quadric is defined by an equation of degree 2 and this has consequences…
• the geometric aspects: a conic of the Euclidean plane can be given a strictly metric definition… and this also has consequences.
Michèle Audin

### Chapter VII. Curves, Envelopes, Evolutes

Abstract
Curves appear in a wide variety of mathematical problems and in very many ways. Here is a list of examples (in a real affine space):
• in kinematics, e.g., as integral curves of vector fields, in other words as solutions of differential equations;
• a variant: as envelopes of families of lines, or of families of other curves;
• they also appear, in space, at the intersection of two surfaces (e.g., of a surface and a plane), therefore, they are useful for the study of surfaces (see Chapter VIII);
• as the solutions of an algebraic equation P( x, y) = 0 (as conics in the plane, for instance), or of several algebraic equations (P1 (x, y, z) = P2 (x, y, z) = 0 in space(1) of dimension 3, etc.).
Michèle Audin

### Chapter VIII. Surfaces in 3-dimensional Space

Abstract
This chapter is an introduction to the local properties of the surfaces in 3-dimensional space. Before coming to (necessarily) heavy definitions, I give a few simple examples of objects which I am sure the reader will agree should be called surfaces: surfaces of revolution, ruled surfaces, etc. I then come to the definitions and to the affine properties, tangent plane and position with respect to the tangent plane, in particular. The last section is devoted to the metric properties of surfaces in a Euclidean space, in particular to the Gauss curvature.
Michèle Audin

### Backmatter

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