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 97 筆
第 2 頁
... point at infinity” and that all such points lie on a single “line at infinity,” we eliminate parallelism. When we admit the new points and the new line into the fold with the same rights and privileges as all the others, we have the ...
... point at infinity” and that all such points lie on a single “line at infinity,” we eliminate parallelism. When we admit the new points and the new line into the fold with the same rights and privileges as all the others, we have the ...
第 3 頁
... points. Two great circles either meet in a pair of antipodal points, are tangent at an equatorial point, or do not meet. Identification of antipodal points converts great circles of the hyperbolic sphere into lines of the hyperbolic ...
... points. Two great circles either meet in a pair of antipodal points, are tangent at an equatorial point, or do not meet. Identification of antipodal points converts great circles of the hyperbolic sphere into lines of the hyperbolic ...
第 9 頁
... points” are one-dimensional subspaces 〈x〉 spanned by nonzero vectors x ∈ V. A module has the structure of a ... points of a geometric line and elements of some number system. When this system is an ordered field (e.g., Q or R), sets ...
... points” are one-dimensional subspaces 〈x〉 spanned by nonzero vectors x ∈ V. A module has the structure of a ... points of a geometric line and elements of some number system. When this system is an ordered field (e.g., Q or R), sets ...
第 10 頁
... point but one of a “chain” corresponds to a unique real number. (If the definition of continuity were modified to allow infinitesimal quantities, one could even identify the points of a line with the “hyperreal” numbers of nonstandard ...
... point but one of a “chain” corresponds to a unique real number. (If the definition of continuity were modified to allow infinitesimal quantities, one could even identify the points of a line with the “hyperreal” numbers of nonstandard ...
第 13 頁
... points can be linked by a path, and every simple closed curve can be shrunk to a point. Having constant curvature means that each of these spaces is homogeneous (any two points are related by a distance-preserving transformation) and ...
... points can be linked by a path, and every simple closed curve can be shrunk to a point. Having constant curvature means that each of these spaces is homogeneous (any two points are related by a distance-preserving transformation) and ...
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 |
其他版本 - 查看全部
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