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. |
搜尋書籍內容
第 1 到 5 筆結果,共 80 筆
第 xii 頁
... corresponding bilinear and quadratic forms. The pseudo-symmetry of real and imaginary is likewise evident in inversive geometry as well as in the metric properties of hyperbolic and elliptic geometry. With coordinates taken as vectors ...
... corresponding bilinear and quadratic forms. The pseudo-symmetry of real and imaginary is likewise evident in inversive geometry as well as in the metric properties of hyperbolic and elliptic geometry. With coordinates taken as vectors ...
第 9 頁
... corresponding to each linear form on a given vector space V is a covector of the dual vector space V.ˇ The annihilator of a vector x ∈ V is the set of covectors ˇu ∈ Vˇ for which the corresponding linear form maps x to 0. If V is a ...
... corresponding to each linear form on a given vector space V is a covector of the dual vector space V.ˇ The annihilator of a vector x ∈ V is the set of covectors ˇu ∈ Vˇ for which the corresponding linear form maps x to 0. If V is a ...
第 12 頁
... corresponding to linear fractional transformations of C∪ {∞}. Besides the mostly continuous groups of transformations of whole spaces, we shall also investigate the discrete symmetry groups of certain geometric figures, such as ...
... corresponding to linear fractional transformations of C∪ {∞}. Besides the mostly continuous groups of transformations of whole spaces, we shall also investigate the discrete symmetry groups of certain geometric figures, such as ...
第 31 頁
... corresponding vertices are concurrent, and perspective from a line if the points in which corresponding sides meet are collinear. Given Axioms I, J, K, L, and M, one can prove the following result, a special case of the celebrated two ...
... corresponding vertices are concurrent, and perspective from a line if the points in which corresponding sides meet are collinear. Given Axioms I, J, K, L, and M, one can prove the following result, a special case of the celebrated two ...
第 32 頁
... corresponding vertices, the points in which corresponding sides meet, and the center and axis of perspectivity form a Desargues configuration 103 of ten points and ten lines, with three lines through each point and three points on each ...
... corresponding vertices, the points in which corresponding sides meet, and the center and axis of perspectivity form a Desargues configuration 103 of ten points and ten lines, with three lines through each point and three points on each ...
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