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 27 筆
第 7 頁
... fixed element of a group G, the mapping x ↦→ axa−1 (“conjugation by a”) is an inner automorphism of G. When transformations of geometric points are expressed as algebraic homomorphisms, successive operations are normally carried out ...
... fixed element of a group G, the mapping x ↦→ axa−1 (“conjugation by a”) is an inner automorphism of G. When transformations of geometric points are expressed as algebraic homomorphisms, successive operations are normally carried out ...
第 14 頁
... fixed point in E2 is a straight line, but in H2 the limit is a horocycle— a “circle” with its center at infinity. Similarly, the locus of points at a constant distance from a given line in E2 is a pair of parallel lines, but in H2 it is ...
... fixed point in E2 is a straight line, but in H2 the limit is a horocycle— a “circle” with its center at infinity. Similarly, the locus of points at a constant distance from a given line in E2 is a pair of parallel lines, but in H2 it is ...
第 18 頁
... 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”) points. When antipodal points are identified, Sn is converted to elliptic n ...
... 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”) points. When antipodal points are identified, Sn is converted to elliptic n ...
第 19 頁
... fixed. The center of On—the subgroup of isometries of Sn−1 that commute with every isometry—is the unit scalar (or “central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by '2 ...
... fixed. The center of On—the subgroup of isometries of Sn−1 that commute with every isometry—is the unit scalar (or “central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by '2 ...
第 23 頁
... fixed “elliptic” inversion of In becomes the central inversion of Sn, which is an inversion in a point—the center of an n-sphere—when Sn is embedded in some (n + 1)-dimensional metric space. A fixed “hyperbolic” inversion of In defining ...
... fixed “elliptic” inversion of In becomes the central inversion of Sn, which is an inversion in a point—the center of an n-sphere—when Sn is embedded in some (n + 1)-dimensional metric space. A fixed “hyperbolic” inversion of In defining ...
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