Geometries
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.} The real projective line, cross ratio, harmonic quadruple, projective scale. \textit{Excursion 4.} Projective scales in the real world. \textit{Chapter 5.} Conics, tangents, polarity, Plücker's \(\mu\). \textit{Excursion 5.} On a photograph five points are clearly identifiable; determine the photographer's position! \textit{Chapter 6.} Application of complex numbers in geometry, the complex projective line \(\mathrm{PG}(1,{\mathbb C})\) and its transformations. The circular points at infinity and orthogonality. \textit{Excursion 6.} The aesthetic of Möbius transformations. \textit{Chapter 7.} Plane Euclidean geometry, similarity, measure of angles (formula of Laguerre), measure of length, the theorems of Ptolemy, Pythagoras, and Miguel, the center of a circle as intersection of the circle tangents at the circular points at infinity, foci. \textit{Excursion 7.} Plane pseudo-Euclidean geometry. \textit{Chapter 8.} The \(3\)-dimensional real projective space \(\mathrm{PG}(3,{\mathbb R})\), points, planes, the Plücker coordinates of a line, join and meet in \(\mathrm{PG}(3,{\mathbb R})\) (the authors use an interesting system to handle indices). \textit{Excursion 8.} Robots. \textit{Chapter 9.} Determinants, configuration of points, projective invariant properties, the Grassmann-Plücker relations. \textit{Excursion 9.} Computer aided proving demonstrated with the theorems of Desargues and Pascal. \textit{Chapter 10.} Introduction to Lie coordinates, problem of Apollonios. \textit{Excursion 10.} Soddy circles (theorem of Descartes), integer Apollonian circle packings. \textit{Chapter 11.} Lie transformations, the connection between Lie and Möbius geometry, Lorentz coordinates and Lorentz quadric, stereographic projection. \textit{Excursion 11.} Steiner's circle chains. \textit{Chapter 12.} Unitary matrices, rotations of \({\mathbb R^3}\), eigen values and eigen vectors, quaternions and their application to rotations of \({\mathbb R^3}\). \textit{Excursion 12.} Octonions, theorem of Milnor and Bott on the paralellizability of spheres, and the hairy ball theorem.
scientific article; zbMATH DE number 5633106