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 84 筆
第 xii 頁
... represented isometrically by a real space of dimension 2n or 2n + 1, and vice versa—a familiar example being the Argand diagram identifying complex numbers with points of the Euclidean plane. Many of the connections between real spaces ...
... represented isometrically by a real space of dimension 2n or 2n + 1, and vice versa—a familiar example being the Argand diagram identifying complex numbers with points of the Euclidean plane. Many of the connections between real spaces ...
第 8 頁
... represented by matrices. Basic geometric properties, such as distances and angles, are defined by means of bilinear forms, functions V × V → F that map pairs of vectors into scalars, preserving linear combinations. The relevant theory ...
... represented by matrices. Basic geometric properties, such as distances and angles, are defined by means of bilinear forms, functions V × V → F that map pairs of vectors into scalars, preserving linear combinations. The relevant theory ...
第 11 頁
... represented by matrices. When coordinate vectors (x) = (x1,...,xn) are regarded as rows (1×n matrices), a linear transformation ·A, determined by an n × n matrix A, maps the point X with coordinates (x) to the point X with coordinates ...
... represented by matrices. When coordinate vectors (x) = (x1,...,xn) are regarded as rows (1×n matrices), a linear transformation ·A, determined by an n × n matrix A, maps the point X with coordinates (x) to the point X with coordinates ...
第 12 頁
... represented by 2 × 2 invertible matrices over F, constituting the projective general linear group PGL2(F). This book is primarily concerned with geometries that can be coordinatized over the real field R, but many results can be ...
... represented by 2 × 2 invertible matrices over F, constituting the projective general linear group PGL2(F). This book is primarily concerned with geometries that can be coordinatized over the real field R, but many results can be ...
第 44 頁
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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