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 6 到 10 筆結果,共 94 筆
第 8 頁
... space F n comprises all lists (x1,...,xn) of n elements of F (the “entries” of the list), with element-by-element addition and scalar multiplication. The vector space F 1 is the field F itself. Given an ordered set [x1,...,xk] of ...
... space F n comprises all lists (x1,...,xn) of n elements of F (the “entries” of the list), with element-by-element addition and scalar multiplication. The vector space F 1 is the field F itself. Given an ordered set [x1,...,xk] of ...
第 12 頁
... n-space for 2 ≤ n ≤ 5 are defined by linear fractional transformations (or 2 × 2 matrices) over rings of real, complex, or quaternionic integers. 1 HOMOGENEOUS SPACES Euclidean geometry is the prototype of a 12 Preliminaries.
... n-space for 2 ≤ n ≤ 5 are defined by linear fractional transformations (or 2 × 2 matrices) over rings of real, complex, or quaternionic integers. 1 HOMOGENEOUS SPACES Euclidean geometry is the prototype of a 12 Preliminaries.
第 13 頁
... spaces can be distinguished by how their subspaces are related. Every isometry can be expressed as the product of reffections, a fact that enables us to classify the different isometries of n-dimensional space. Complex conjugation ...
... spaces can be distinguished by how their subspaces are related. Every isometry can be expressed as the product of reffections, a fact that enables us to classify the different isometries of n-dimensional space. Complex conjugation ...
第 14 頁
... spaces. Here we note a few of their distinctive properties. Each n-dimensional real metric space contains various lowerdimensional subspaces. A k-dimensional subspace of Euclidean, hyperbolic, or elliptic n-space (k < n) is a k-plane Ek ...
... spaces. Here we note a few of their distinctive properties. Each n-dimensional real metric space contains various lowerdimensional subspaces. A k-dimensional subspace of Euclidean, hyperbolic, or elliptic n-space (k < n) is a k-plane Ek ...
第 16 頁
... n-space the limit of a k-sphere Sk (k < n) is a k-plane Ek, while in hyperbolic n-space it is a k-horosphere, isometric to Euclidean k-space. Hyperbolic k-space can be embedded in hyperbolic n-space (k < n) not only as a k-plane Hk but ...
... n-space the limit of a k-sphere Sk (k < n) is a k-plane Ek, while in hyperbolic n-space it is a k-horosphere, isometric to Euclidean k-space. Hyperbolic k-space can be embedded in hyperbolic n-space (k < n) not only as a k-plane Hk but ...
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