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 79 筆
第 viii 頁
... . 162 7.3 Summary of Complex Spaces .................... 166 8 Complex Collineation Groups ........................ 168 8.1 Linear and Affine Transformations ................ 168 8.2 Projective Transformations ...................... 175 ...
... . 162 7.3 Summary of Complex Spaces .................... 166 8 Complex Collineation Groups ........................ 168 8.1 Linear and Affine Transformations ................ 168 8.2 Projective Transformations ...................... 175 ...
第 xii 頁
... complex projective space and coordinatized over the complex field. A unitary n-space may be represented isometrically by a real space of dimension 2n or 2n + 1, and vice versa—a familiar example being the Argand diagram identifying complex ...
... complex projective space and coordinatized over the complex field. A unitary n-space may be represented isometrically by a real space of dimension 2n or 2n + 1, and vice versa—a familiar example being the Argand diagram identifying complex ...
第 10 頁
... complex geometries apart from others is continuity, which essentially means that no points are “missing” from a line. The notion of continuity is implicit in the theory of proportion developed by Eudoxus (fourth century BC) and ...
... complex geometries apart from others is continuity, which essentially means that no points are “missing” from a line. The notion of continuity is implicit in the theory of proportion developed by Eudoxus (fourth century BC) and ...
第 12 頁
... complex field C and other systems, including the noncommutative division ring (“skew-field”) of quaternions. Also, real spaces may be represented by complex or quaternionic models. For example, the real inversive sphere I 2 can be ...
... complex field C and other systems, including the noncommutative division ring (“skew-field”) of quaternions. Also, real spaces may be represented by complex or quaternionic models. For example, the real inversive sphere I 2 can be ...
第 13 頁
... Complex conjugation allows a real-valued distance function to be defined on spaces coordinatized over the complex field, and each real metric space has a unitary counterpart. 1.1 REAL METRIC SPACES The classical real metric spaces are ...
... Complex conjugation allows a real-valued distance function to be defined on spaces coordinatized over the complex field, and each real metric space has a unitary counterpart. 1.1 REAL METRIC SPACES The classical real metric spaces are ...
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