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 45 筆
第 2 頁
... Extended lines and ordinary circles together form a set of “inversive circles” on the real inversive sphere I2. Any three points lie on a unique inversive circle; points lying on the same circle are concyclic. Two circles may meet in ...
... Extended lines and ordinary circles together form a set of “inversive circles” on the real inversive sphere I2. Any three points lie on a unique inversive circle; points lying on the same circle are concyclic. Two circles may meet in ...
第 9 頁
... extended to the (n − 1)-dimensional projective space PV whose “points” are one-dimensional subspaces 〈x〉 spanned by nonzero vectors x ∈ V. A module has the structure of a vector space except that scalars are only required to belong ...
... extended to the (n − 1)-dimensional projective space PV whose “points” are one-dimensional subspaces 〈x〉 spanned by nonzero vectors x ∈ V. A module has the structure of a vector space except that scalars are only required to belong ...
第 11 頁
... extended value ∞ (an extra element that behaves like the reciprocal of 0). A “projectivity” of FP1 can be expressed as a linear fractional transformation of the extended field F∪ {∞}, defined for given field elements a, b, c, d (ad ...
... extended value ∞ (an extra element that behaves like the reciprocal of 0). A “projectivity” of FP1 can be expressed as a linear fractional transformation of the extended field F∪ {∞}, defined for given field elements a, b, c, d (ad ...
第 16 頁
... extended by adjoining to the ordinary points of each line one absolute point (or “point at infinity”), the resulting extended Euclidean space being denoted En. ̄ All the absolute points of En ̄ lie in an absolute hyperplane, and each ...
... extended by adjoining to the ordinary points of each line one absolute point (or “point at infinity”), the resulting extended Euclidean space being denoted En. ̄ All the absolute points of En ̄ lie in an absolute hyperplane, and each ...
第 18 頁
... extended hyperbolic n-space Hn ̄ (Johnson 1981, pp. 452–454). 1.2 ISOMETRIES According to the Erlanger Programm of Klein, a geometry is characterized by the group of transformations that preserve its essential properties. Each of the ...
... extended hyperbolic n-space Hn ̄ (Johnson 1981, pp. 452–454). 1.2 ISOMETRIES According to the Erlanger Programm of Klein, a geometry is characterized by the group of transformations that preserve its essential properties. Each of the ...
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