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 66 筆
第 i 頁
... 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 ...
... 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 ...
第 ii 頁
... preserve their essential characteristic features. In this process, he follows the spirit of Felix Klein and his Erlangen Program, first enunciated in 1870. Once again, a specific personal contribution to this achievement is his facility ...
... preserve their essential characteristic features. In this process, he follows the spirit of Felix Klein and his Erlangen Program, first enunciated in 1870. Once again, a specific personal contribution to this achievement is his facility ...
第 5 頁
... preserving the fundamental properties of the geometry is represented by a particular type of invertible matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates and matrix entries may be real ...
... preserving the fundamental properties of the geometry is represented by a particular type of invertible matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates and matrix entries may be real ...
第 6 頁
... preserves the system operation(s), carrying sums or products in the domain U into sums or products in the codomain V. The kernel Ker T is the set of elements in U that are mapped into the identity element of V (the zero element in the ...
... preserves the system operation(s), carrying sums or products in the domain U into sums or products in the codomain V. The kernel Ker T is the set of elements in U that are mapped into the identity element of V (the zero element in the ...
第 8 頁
... preserving vector sums and scalar multiples, is a linear transformation. (When scalars are written on the left, linear transformations go on the right, and vice versa.) If V is a vector space over F, a linear transformation V → F is a ...
... preserving vector sums and scalar multiples, is a linear transformation. (When scalars are written on the left, linear transformations go on the right, and vice versa.) If V is a vector space over F, a linear transformation V → F is a ...
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