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 54 筆
第 xi 頁
... meet in at most one point. But spherical space is “circular”: two great circles on a sphere always meet in a pair of antipodal points. Each of these spaces can be coordinatized over the real field, and each of them can xi Preface page.
... meet in at most one point. But spherical space is “circular”: two great circles on a sphere always meet in a pair of antipodal points. Each of these spaces can be coordinatized over the real field, and each of them can xi Preface page.
第 3 頁
... pairs, and the role of lines is played by great circles; any two nonantipodal points lie on a unique great circle, and any two great circles meet in a pair of antipodal points. When antipodal points are identified, great circles of the ...
... pairs, and the role of lines is played by great circles; any two nonantipodal points lie on a unique great circle, and any two great circles meet in a pair of antipodal points. When antipodal points are identified, great circles of the ...
第 8 頁
... defined by means of bilinear forms, functions V × V → F that map pairs of vectors into scalars, preserving linear combinations. The relevant theory of finite-dimensional vector spaces will be developed beginning 8 Preliminaries.
... defined by means of bilinear forms, functions V × V → F that map pairs of vectors into scalars, preserving linear combinations. The relevant theory of finite-dimensional vector spaces will be developed beginning 8 Preliminaries.
第 10 頁
... pair of points separating another pair. The order relation can be used to define line segments, rays, and the like. The property that sets real and complex geometries apart from others is continuity, which essentially means that no ...
... pair of points separating another pair. The order relation can be used to define line segments, rays, and the like. The property that sets real and complex geometries apart from others is continuity, which essentially means that no ...
第 13 頁
... pair of points is a nonnegative real number, so that lengths and angles can be measured in a consistent way. In a homogeneous space, any point can be taken to any other point by a distance-preserving transformation, or isometry ...
... pair of points is a nonnegative real number, so that lengths and angles can be measured in a consistent way. In a homogeneous space, any point can be taken to any other point by a distance-preserving transformation, or isometry ...
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