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. |
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第 1 到 5 筆結果,共 73 筆
第 ix 頁
... Coxeter Groups ....................... 246 11.6 Subgroups and Extensions ...................... 254 Euclidean ... Group .................... 337 14.3 The Eisenstein Modular Group ................... 342 Quaternionic Modular Groups ...
... Coxeter Groups ....................... 246 11.6 Subgroups and Extensions ...................... 254 Euclidean ... Group .................... 337 14.3 The Eisenstein Modular Group ................... 342 Quaternionic Modular Groups ...
第 xiii 頁
... group is placed in an algebraic context, so that its relationship to other groups may be readily seen. While there ... Coxeter groups—are of particular importance. Finite and infinite symmetry groups are subgroups or extensions of spherical ...
... group is placed in an algebraic context, so that its relationship to other groups may be readily seen. While there ... Coxeter groups—are of particular importance. Finite and infinite symmetry groups are subgroups or extensions of spherical ...
第 xiv 頁
... Coxeter, as well as relevant works by contemporary authors. In any case ... group theory. I also take for granted the reader's familiarity with the ... Coxeter, who in the course of his long life did much to rescue geometry from oblivion ...
... Coxeter, as well as relevant works by contemporary authors. In any case ... group theory. I also take for granted the reader's familiarity with the ... Coxeter, who in the course of his long life did much to rescue geometry from oblivion ...
第 20 頁
... group of isometries is its orbit. If every point has a neighborhood that ... (Coxeter 1988, p. 43). Thus the identity is of type 0 and a reffection is of ... (Coxeter 1942, pp. 114, 130). In an orientable space, an isometry of type l is ...
... group of isometries is its orbit. If every point has a neighborhood that ... (Coxeter 1988, p. 43). Thus the identity is of type 0 and a reffection is of ... (Coxeter 1942, pp. 114, 130). In an orientable space, an isometry of type l is ...
第 198 頁
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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