Geometries
type: Item
instance of: scholarly article
title: Geometries
zbMATH Open document ID: 1181.00003
DOI: 10.1007/978-3-642-02530-3
author: Jürgen Richter-Gebert
author: Thorsten Orendt
published in: Springer-Lehrbuch
publication date: 2009-11-13T00:00:00Z
review text: The reader of this book is assumed to have a basic knowledge of linear algebra (vector space, matrices, cross product, determinants, eigen vectors) and of complex numbers (definition, polar coordinates). The aim of the book is a very pragmatic introduction to the classic and modern approaches to geometry. Two tasks lead the presentation: 1. Calculation with geometric objects. 2. Development of a computer program for geometry [cf. \textit{J. Richter-Gebert} and \textit{U. Kortenkamp}, Die interaktive Geometrie-Software Cinderella. Version 1. 2. Einzelplatzversion. Berlin: Springer. CD-ROM, Handbuch x (1999; Zbl 0969.51001); \url{http://www.cinderella.de}]. The underlying structures are either finite dimensional real projective spaces or the complex projective line whereas abstract theory of projective planes and spaces is avoided completely. By using determinants in nearly all formulas the authors achieve a very unified presentation. The book addresses students of mathematics, informatics and physics after their first year at university and, of course, docents. Each of the well--motivated twelve chapters of the book is accompanied by exercises and an \textit{excursion} where some light is shed on advanced and aesthetical aspects or on applications. The book contains numerous excellent illustrations and finishes with a survey of references being useful for deepening the presented topics and an index. \textit{Chapter 1} deals with the real projective plane \(\mathrm{PG}(2,{\mathbb R})\), homogeneous coordinates, incidence, the operations join and meet and the fact that \(\mathrm{PG}(2,{\mathbb R})\) is not orientable. \textit{Excursion 1} exhibits orientable and non-orientable surfaces. \textit{Chapter 2.} Euclidean, affine, projective transformations. \textit{Excursion 2.} Reconstruction of plane figures from their photographs. \textit{Chapter 3.} Duality of \(\mathrm{PG}(2,{\mathbb R})\). \textit{Excursion 3.} Symmetries of the Pappos configuration. \textit{Chapter 4…
reviewed by: Rolf Riesinger
Mathematics Subject Classification ID: 00A05
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51B10
Mathematics Subject Classification ID: 51B25
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51N15
Mathematics Subject Classification ID: 65Z05
zbMATH DE Number: 5633106
zbMATH Keywords: homogeneous coordinates
zbMATH Keywords: projective scale
zbMATH Keywords: circular points at infinity
zbMATH Keywords: isotropic directions
zbMATH Keywords: Plücker's line coordinates
zbMATH Keywords: Grassmann-Plücker relations
zbMATH Keywords: circle geometry
zbMATH Keywords: polynom of determinants
zbMATH Keywords: Lorentz transformation
zbMATH Keywords: quaternion
zbMATH Keywords: octonion
describes a project that uses: Cinderella
describes a project that uses: jReality
describes a project that uses: PovRay
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-02530-3
OpenAlex ID: W4254160295
description: scientific article; zbMATH DE number 5633106