Geometries and TransformationsCambridge University Press, 2018年6月7日 Euclidean and other geometries are distinguished by the transformations that preserve their essential properties. Using linear algebra and transformation groups, this book provides a readable exposition of how these classical geometries are both differentiated and connected. Following Cayley and Klein, the book builds on projective and inversive geometry to construct 'linear' and 'circular' geometries, including classical real metric spaces like Euclidean, hyperbolic, elliptic, and spherical, as well as their unitary counterparts. The first part of the book deals with the foundations and general properties of the various kinds of geometries. The latter part studies discrete-geometric structures and their symmetries in various spaces. Written for graduate students, the book includes numerous exercises and covers both classical results and new research in the field. An understanding of analytic geometry, linear algebra, and elementary group theory is assumed. |
搜尋書籍內容
第 1 到 5 筆結果,共 33 筆
第 xii 頁
... polarity in projective n-space Pn or one of its hyperplanes. Fixing a central inversion in inversive n-space, realized as an oval n-quadric in Pn+1, similarly produces one or another kind of circular metric space. Each of these real ...
... polarity in projective n-space Pn or one of its hyperplanes. Fixing a central inversion in inversive n-space, realized as an oval n-quadric in Pn+1, similarly produces one or another kind of circular metric space. Each of these real ...
第 9 頁
... polarity provided that x0 ˇy whenever y0 ˇx, and vectors x and y are said to be conjugate in the polarity. These concepts can be extended to the (n − 1)-dimensional projective space PV whose “points” are one-dimensional subspaces 〈x〉 ...
... polarity provided that x0 ˇy whenever y0 ˇx, and vectors x and y are said to be conjugate in the polarity. These concepts can be extended to the (n − 1)-dimensional projective space PV whose “points” are one-dimensional subspaces 〈x〉 ...
第 17 頁
... polarity that pairs each ordinary, absolute, or ideal point with an ideal, absolute, or ordinary hyperplane. Indeed, as shown by Cayley (1859) and Klein (1871, 1873), each of the non-Euclidean n-spaces ePn and Hn (as contained in hPn) ...
... polarity that pairs each ordinary, absolute, or ideal point with an ideal, absolute, or ordinary hyperplane. Indeed, as shown by Cayley (1859) and Klein (1871, 1873), each of the non-Euclidean n-spaces ePn and Hn (as contained in hPn) ...
第 18 頁
... polarity in the (projective) hyperplane at infinity. For both ePn and En, the polarity is elliptic; i.e., there are no self-conjugate points. For hPn, the polarity is hyperbolic: the locus of self-conjugate points— the absolute points ...
... polarity in the (projective) hyperplane at infinity. For both ePn and En, the polarity is elliptic; i.e., there are no self-conjugate points. For hPn, the polarity is hyperbolic: the locus of self-conjugate points— the absolute points ...
第 24 頁
... in Chapter 7. Self-conjugate points of a polarity correspond to solutions of a quadratic equation. By the Fundamental Theorem of Algebra, the complex field C is algebraically closed, so that there are 24 Homogeneous Spaces Unitary Spaces.
... in Chapter 7. Self-conjugate points of a polarity correspond to solutions of a quadratic equation. By the Fundamental Theorem of Algebra, the complex field C is algebraically closed, so that there are 24 Homogeneous Spaces Unitary Spaces.
內容
1 | |
13 | |
27 | |
Circular Geometries | 57 |
Real Collineation Groups | 87 |
Equiareal Collineations | 113 |
Real Isometry Groups | 138 |
Complex Spaces | 157 |
Complex Collineation Groups | 168 |
Circularities and Concatenations | 183 |
Unitary Isometry Groups | 203 |
Finite Symmetry Groups | 223 |
Tables | 390 |
List of Symbols | 406 |
Index | 425 |
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常見字詞
affine angle associated Axiom called central circle collineation column commutator complex contains coordinates corresponding Coxeter diagrams Coxeter group defined determinant direct distance dual elements elliptic entries equal Euclidean EXERCISES expressed extended field Figure Find finite fixed follows four fractional transformations fundamental region geometry given half-turn honeycomb hyperbolic hyperplane hypersphere induces infinite integers inversive isometry isomorphic lattice length linear group mapping matrix meet multiplication n-space nonzero normal obtain operation ordinary orthogonal orthogonal matrix pairs parallel period plane points polarity positive preserves projective properties quaternionic ratios reffections regular represented respective ring rotation satisfying the relations scalar separated Show sides similarity space sphere spherical subgroup of index symbol symmetry group symplectic taking tions transformation translation triangle unique unit unitary vector vector space vertices