Geometry. Transl. from the French
type: Item
instance of: scholarly article
title: Geometry. Transl. from the French
zbMATH Open document ID: 1043.51001
DOI: 10.1007/978-3-642-56127-6
author: Michèle Audin
published in: Universitext
publication date: 2002-10-30T00:00:00Z
review text: 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, t…
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 51Nxx
Mathematics Subject Classification ID: 53-01
zbMATH DE Number: 1821645
zbMATH Keywords: affine geometry
zbMATH Keywords: Euclidean geometry
zbMATH Keywords: Euclidean geometry in the plane
zbMATH Keywords: Euclidean geometry in space
zbMATH Keywords: projective geometry
zbMATH Keywords: conics
zbMATH Keywords: quadrics
zbMATH Keywords: curves
zbMATH Keywords: envelopes
zbMATH Keywords: evolutes
zbMATH Keywords: surfaces in 3-dimensional space
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-56127-6
OpenAlex ID: W4238547711
description: scientific article; zbMATH DE number 1821645