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 44 筆
第 xii 頁
... central inversion in inversive n-space, realized as an oval n-quadric in Pn+1, similarly produces one or another kind of circular metric space. Each of these real metric spaces has a unitary counterpart that can be derived analogously ...
... central inversion in inversive n-space, realized as an oval n-quadric in Pn+1, similarly produces one or another kind of circular metric space. Each of these real metric spaces has a unitary counterpart that can be derived analogously ...
第 4 頁
... central quotient group. If a and b are two elements of a group G, the element a−1b−1ab is their commutator; this differs from the identity precisely when ab = ba. The set of commutators generates a normal 4 Preliminaries.
... central quotient group. If a and b are two elements of a group G, the element a−1b−1ab is their commutator; this differs from the identity precisely when ab = ba. The set of commutators generates a normal 4 Preliminaries.
第 11 頁
... central quotient group, the projective general linear group PGLn. In a variation of this procedure, each point of a projective line FP1 over a field F can be identified uniquely either with some element x ∈ F or with the extended value ...
... central quotient group, the projective general linear group PGLn. In a variation of this procedure, each point of a projective line FP1 over a field F can be identified uniquely either with some element x ∈ F or with the extended value ...
第 18 頁
... central n-sphere, with a metric. If the central inversion so defined has no fixed points, this is the elliptic n-sphere Sn. Otherwise, we have the hyperbolic n-sphere Sn, ̈ with an “equatorial” (n−1)-sphere of fixed (“self-antipodal ...
... central n-sphere, with a metric. If the central inversion so defined has no fixed points, this is the elliptic n-sphere Sn. Otherwise, we have the hyperbolic n-sphere Sn, ̈ with an “equatorial” (n−1)-sphere of fixed (“self-antipodal ...
第 19 頁
... central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by '2'. Elliptic (n − 1)-space ePn−1 = Sn−1/2 results from the identification of antipodal points of Sn−1, and its isometry ...
... central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by '2'. Elliptic (n − 1)-space ePn−1 = Sn−1/2 results from the identification of antipodal points of Sn−1, and its 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