Geometry. Transl. from the French
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. In Ch. 1, Affine geometry, the author explains that linear algebra is useful for elementary geometry (after all, this is where it comes from) and to show ``genuine'' geometry: triangles, spheres, polyhedra, angles at the circumference, inversions, parabolas. In Ch. II, Euclidean geometry, feneralities, all the vector spaces are defined over the field \(\mathbb R\) of real numbers. The spaces under consideration all have finite dimension. Tools allowing one to measure distances are introduced. In Ch. III, Euclidean geometry in the plane, there are plane isometries, triangles and angles at the circumference, similarities, inversions and even pencils of circles. But there is also a discussion of what an angle is and how to measure it. The proofs are very simple but the statements and their precision are subtle and important. In Ch. IV, Euclidean geometry in space, everything takes place in a Euclidean (affine or vector) space of dimension 3. After having investigated the group of isometries and recalled a few facts about the vector product, the author attacks spheres and especially spherical triangles, the triangles drawn by great circles on the sphere. Having computed the area of these triangles, the Euler formula for convex polyhedra and the list of regular polyhedra are deduced. In Ch. V, Projective geometry, all spaces, affine, linear and projective are defined over a field \(\mathbb K\). The reader is allowed to think that this field is \(\mathbb R\) or \(\mathbb C\). In Ch. VI, Conics and quadrics, the author tries to keep a balance between the algebraic aspects: a quadric is defined by an equation of degree 2 (and this has consequences), and the geometric aspects: a conic of the Euclidean plane can be given a strictly metric definition (and this also has consequences). The first two sections are elementary and devoted respectively to these two aspects of conics and quadrics. For benefit of more advanced readers, the author then comes to projective conics and quadrics. The properties related to the algebraic definition, mainly polarity and duality, are investigated. The relation to homographies, around Pascal's theorem, is sketched. It is also shown that the affine \textit{and} metric properties of affine conics may be deduced from the projective approach. In the last section, it is shown that the results about pencils of circles and inversions can be expressed very simply by looking at the lines intersecting a suitable quadric in a (suitable) space of dimension 3. For the convenience of the reader, an appendix on quadratic forms and a few ``stretching'' exercises is included. In Ch. VII, Curves, envelopes, evolutes, the author looks first at envelope problems. The reader is reminded of a few results on the curvature of plane curves, in order to come back to envelopes in the framework of evolutes. Ch. VIII is an introduction to the local properties of the ``Surfaces in 3-dimensional space''. Affine properties of surfaces, tangent plane and position with respect to the tangent plane, in particular, are given. The last section is devoted to the metric properties of surfaces in a Euclidean space, in particular to the Gauss curvature. Although there is no global result in this chapter, it contains enough examples and applications not to be boring. Everything is presented clearly and rigorously. 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.
scientific article; zbMATH DE number 1821645