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 69 筆
第 19 頁
... orthogonal group On ∼= En/Tn, induced by the isometries of En that leave a given point 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 ...
... orthogonal group On ∼= En/Tn, induced by the isometries of En that leave a given point 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 ...
第 21 頁
... orthogonal to some ordinary line is a translation along the line, the axis of the translation, through twice the distance between the mirrors. When the mirrors are coincident, the translation reduces to the identity. Otherwise, in En (n ...
... orthogonal to some ordinary line is a translation along the line, the axis of the translation, through twice the distance between the mirrors. When the mirrors are coincident, the translation reduces to the identity. Otherwise, in En (n ...
第 22 頁
... orthogonal reffection. A striaflection (or “parallel reffection”) in Hn (n ≥ 3) is the product of a striation and a reffection. A rotary translation, the product in En or Hn (n ≥ 3) of a rotation and an orthogonal translation, may be ...
... orthogonal reffection. A striaflection (or “parallel reffection”) in Hn (n ≥ 3) is the product of a striation and a reffection. A rotary translation, the product in En or Hn (n ≥ 3) of a rotation and an orthogonal translation, may be ...
第 23 頁
... orthogonal great hyperspheres of Sn is the central inversion. The product of reffections in n + 1 mutually orthogonal hyperplanes of ePn is the identity. Some writers call all involutory isometries “reffections.” The term “inversion” is ...
... orthogonal great hyperspheres of Sn is the central inversion. The product of reffections in n + 1 mutually orthogonal hyperplanes of ePn is the identity. Some writers call all involutory isometries “reffections.” The term “inversion” is ...
第 24 頁
... orthogonal great circles of S2 is a half-turn. 2. ⊣The product of two half-turns in E2 is a translation. 3. ⊣ The set of all translations and half-turns in E2 forms a group. 4. ⊣ Every isometry in S2 is a reffection, a rotation, or ...
... orthogonal great circles of S2 is a half-turn. 2. ⊣The product of two half-turns in E2 is a translation. 3. ⊣ The set of all translations and half-turns in E2 forms a group. 4. ⊣ Every isometry in S2 is a reffection, a rotation, or ...
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