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 6 到 10 筆結果,共 81 筆
第 20 頁
... reffections. Let us say that an isometry of n-space is of type l if it can be expressed as the product of l, but no fewer, reffections (Coxeter 1988, p. 43). Thus the identity is of type 0 and a reffection is of type 1. By a famous ...
... reffections. Let us say that an isometry of n-space is of type l if it can be expressed as the product of l, but no fewer, reffections (Coxeter 1988, p. 43). Thus the identity is of type 0 and a reffection is of type 1. By a famous ...
第 21 頁
... reffections in parallel mirrors is a striation, essentially a “pararotation” about the common absolute (n − 2)-plane of the two mirrors. When the mirrors coincide, the transformation is just the identity. Except for this, every ...
... reffections in parallel mirrors is a striation, essentially a “pararotation” about the common absolute (n − 2)-plane of the two mirrors. When the mirrors coincide, the transformation is just the identity. Except for this, every ...
第 22 頁
... Reffection S, E, H 1 1 Translation E, H 2 1 Striation H 2 2 Rotation S, E, H 2 2 Transffection E, H 3 2 Striaffection H 3 3 Rotary reffection S, E, H 3 3 Rotary translation E, H 4 3 Rotary striation H 4 4 Double rotation S, E, H 4 4 ...
... Reffection S, E, H 1 1 Translation E, H 2 1 Striation H 2 2 Rotation S, E, H 2 2 Transffection E, H 3 2 Striaffection H 3 3 Rotary reffection S, E, H 3 3 Rotary translation E, H 4 3 Rotary striation H 4 4 Double rotation S, E, H 4 4 ...
第 23 頁
... reffections in l mutually orthogonal hyperplanes or great hyperspheres of n-space (1 ≤ l ≤ n) is an involutory ... reffections in n + 1 mutually orthogonal great hyperspheres of Sn is the central inversion. The product of reffections ...
... reffections in l mutually orthogonal hyperplanes or great hyperspheres of n-space (1 ≤ l ≤ n) is an involutory ... reffections in n + 1 mutually orthogonal great hyperspheres of Sn is the central inversion. The product of reffections ...
第 24 頁
... reffections in two perpendicular lines of E2 or H2 or two orthogonal great circles of S2 is a half-turn. 2. ⊣The ... reffection, a rotation, or the commutative product of a reffection and a rotation. 5. ⊣ Every isometry in E3 is either ...
... reffections in two perpendicular lines of E2 or H2 or two orthogonal great circles of S2 is a half-turn. 2. ⊣The ... reffection, a rotation, or the commutative product of a reffection and a rotation. 5. ⊣ Every isometry in E3 is either ...
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