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 51 筆
第 1 頁
... can be measured with the aid of an arbitrarily chosen unit of length. Right angles provide a standard for angular measure. The Euclidean parallel postulate is equivalent to the assertion that through any point 1 Preliminaries.
... can be measured with the aid of an arbitrarily chosen unit of length. Right angles provide a standard for angular measure. The Euclidean parallel postulate is equivalent to the assertion that through any point 1 Preliminaries.
第 2 頁
... parallel lines. Nevertheless, areas can still be determined. Up to size, all triangles are equivalent, as are all parallelograms; there is no such thing as a right triangle or a square. Conics can be distinguished only as ellipses ...
... parallel lines. Nevertheless, areas can still be determined. Up to size, all triangles are equivalent, as are all parallelograms; there is no such thing as a right triangle or a square. Conics can be distinguished only as ellipses ...
第 3 頁
... parallel postulate was eventually shown to be independent of the other assumptions governing the Euclidean plane. Replacing it with the contrary hypothesis—that through any point not on a given line there is more than one line not ...
... parallel postulate was eventually shown to be independent of the other assumptions governing the Euclidean plane. Replacing it with the contrary hypothesis—that through any point not on a given line there is more than one line not ...
第 14 頁
... parallel, and parallel lines in E2 are everywhere equidistant (Figure 1.1b). In the hyperbolic plane H2, two nonintersecting lines are either parallel or diverging (“ultraparallel”). Parallel lines in H2 are asymptotic, converging in ...
... parallel, and parallel lines in E2 are everywhere equidistant (Figure 1.1b). In the hyperbolic plane H2, two nonintersecting lines are either parallel or diverging (“ultraparallel”). Parallel lines in H2 are asymptotic, converging in ...
第 15 頁
... parallel postulate, the geometry of a sphere whose radius increases without bound becomes Euclidean in the limit (Stäckel 1901; cf. Bonola 1906, §30; Sommerville 1914, p. 15;. Figure 1.1a Great circles in S2 Figure 1.1c Parallel lines in ...
... parallel postulate, the geometry of a sphere whose radius increases without bound becomes Euclidean in the limit (Stäckel 1901; cf. Bonola 1906, §30; Sommerville 1914, p. 15;. Figure 1.1a Great circles in S2 Figure 1.1c Parallel lines in ...
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