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 94 筆
第 xii 頁
... coordinates of a projective space we may construct coordinates for a metric space derived from it, in terms of which properties of the space may be described and distances and angles measured. Many of the formulas obtained exhibit an ...
... coordinates of a projective space we may construct coordinates for a metric space derived from it, in terms of which properties of the space may be described and distances and angles measured. Many of the formulas obtained exhibit an ...
第 xv 頁
... and George Olshevsky. And I owe many thanks to my editors and the production staff at Cambridge, especially Katie Leach, Adam Kratoska, and Mark Fox. PRELIMINARIES The invention of coordinates by Pierre de Fermat (1601–1665) Preface xv.
... and George Olshevsky. And I owe many thanks to my editors and the production staff at Cambridge, especially Katie Leach, Adam Kratoska, and Mark Fox. PRELIMINARIES The invention of coordinates by Pierre de Fermat (1601–1665) Preface xv.
第 5 頁
... coordinates, each transformation preserving the fundamental properties of the geometry is represented by a particular type of invertible matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates ...
... coordinates, each transformation preserving the fundamental properties of the geometry is represented by a particular type of invertible matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates ...
第 8 頁
... coordinates of points and hyperplanes function as row and column vectors, and geometric operations—expressed algebraically as linear transformations of coordinates—are represented by matrices. Basic geometric properties, such as ...
... coordinates of points and hyperplanes function as row and column vectors, and geometric operations—expressed algebraically as linear transformations of coordinates—are represented by matrices. Basic geometric properties, such as ...
第 11 頁
... coordinates (x) to the point X with coordinates (x)A, and geometric transformation groups correspond to multiplicative groups of matrices. As a rule, the groups in question turn out to be subgroups of the general linear group GLn of ...
... coordinates (x) to the point X with coordinates (x)A, and geometric transformation groups correspond to multiplicative groups of matrices. As a rule, the groups in question turn out to be subgroups of the general linear group GLn of ...
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