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 83 筆
第 5 頁
... respectively, the rotation group and the full symmetry group (including reffections) of a regular p-gon. A transformation is a permutation of the elements of an arbitrary set—e.g., some or all of the points of a space—or a mapping of ...
... respectively, the rotation group and the full symmetry group (including reffections) of a regular p-gon. A transformation is a permutation of the elements of an arbitrary set—e.g., some or all of the points of a space—or a mapping of ...
第 13 頁
... respectively by Sn, En, Hn, and ePn. For n ≥ 1 such geometries are n-manifolds (locally Euclidean topological spaces); for n ≥ 2 the spaces Sn, En, and Hn are simply connected—any two points can be linked by a path, and every simple ...
... respectively by Sn, En, Hn, and ePn. For n ≥ 1 such geometries are n-manifolds (locally Euclidean topological spaces); for n ≥ 2 the spaces Sn, En, and Hn are simply connected—any two points can be linked by a path, and every simple ...
第 17 頁
... respectively hyperbolic, co-Euclidean, or elliptic. The points and subspaces of an ordinary k-plane hPk may themselves be ordinary, absolute, or ideal. An absolute k-plane Eˇk (dual to Euclidean k-space) has a bundle of absolute ...
... respectively hyperbolic, co-Euclidean, or elliptic. The points and subspaces of an ordinary k-plane hPk may themselves be ordinary, absolute, or ideal. An absolute k-plane Eˇk (dual to Euclidean k-space) has a bundle of absolute ...
第 18 頁
... respectively to points X and Y keeps the distance between them the same: |XY |=|XY|. A similarity of scale μ > 0 has |XY | = μ|XY| for every pair of points. The isometry group of En is the Euclidean group En, a normal subgroup of the ...
... respectively to points X and Y keeps the distance between them the same: |XY |=|XY|. A similarity of scale μ > 0 has |XY | = μ|XY| for every pair of points. The isometry group of En is the Euclidean group En, a normal subgroup of the ...
第 31 頁
... respective points A, B, C, and let the nonconcurrent sides QR, RP, PQ meet Oˇ in the respective points D, E, F. Then we call the ordered hexad (ABC, DEF) a quadrangular set and write ABC ∧ DEF. The points of the set need not all be ...
... respective points A, B, C, and let the nonconcurrent sides QR, RP, PQ meet Oˇ in the respective points D, E, F. Then we call the ordered hexad (ABC, DEF) a quadrangular set and write ABC ∧ DEF. The points of the set need not all be ...
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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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