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 94 筆
第 vii 頁
... Projective n-Space ............................ 39 2.3 Elliptic and Euclidean Geometry .................. 46 2.4 ... Projective Collineations ........................ 102 10 11 4.5 Projective Correlations ......................... 106 ...
... Projective n-Space ............................ 39 2.3 Elliptic and Euclidean Geometry .................. 46 2.4 ... Projective Collineations ........................ 102 10 11 4.5 Projective Correlations ......................... 106 ...
第 viii 頁
... Projective Transformations ...................... 175 8.3 Antiprojective Transformations ................... 178 8.4 Subgroups and Quotient Groups .................. 180 9 Circularities and Concatenations ...................... 183 9.1 ...
... Projective Transformations ...................... 175 8.3 Antiprojective Transformations ................... 178 8.4 Subgroups and Quotient Groups .................. 180 9 Circularities and Concatenations ...................... 183 9.1 ...
第 xi 頁
... projective space. As first demonstrated by von Staudt in 1857, projective spaces have an intrinsic algebraic structure, which is manifested when they are suitably coordinatized. While synthetic methods can still be employed, geometric ...
... projective space. As first demonstrated by von Staudt in 1857, projective spaces have an intrinsic algebraic structure, which is manifested when they are suitably coordinatized. While synthetic methods can still be employed, geometric ...
第 xii 頁
... projective space. For the three linear geometries, a metric can be induced by specializing an absolute polarity in projective n-space Pn or one of its hyperplanes. Fixing a central inversion in inversive n-space, realized as an oval n ...
... projective space. For the three linear geometries, a metric can be induced by specializing an absolute polarity in projective n-space Pn or one of its hyperplanes. Fixing a central inversion in inversive n-space, realized as an oval n ...
第 xiii 頁
... projective geometry, or non-Euclidean geometry, nor is it an attempt to synthesize some combination of those subjects in the manner of Baer (1952) or Onishchik & Sulanke (2006). My objective is rather to explain how all these things fit ...
... projective geometry, or non-Euclidean geometry, nor is it an attempt to synthesize some combination of those subjects in the manner of Baer (1952) or Onishchik & Sulanke (2006). My objective is rather to explain how all these things fit ...
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