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 28 筆
第 4 頁
... normal (or “selfconjugate”) if for any element h ∈ H its conjugate ghg−1 by any element g ∈ G is in H, i.e., if gHg−1 ⊆ H for all g∈ G; we write this as H G. If H is a normal subgroup of G, then for every g ∈ G, the left coset gH ...
... normal (or “selfconjugate”) if for any element h ∈ H its conjugate ghg−1 by any element g ∈ G is in H, i.e., if gHg−1 ⊆ H for all g∈ G; we write this as H G. If H is a normal subgroup of G, then for every g ∈ G, the left coset gH ...
第 5 頁
... normal subgroups of G. A group G may be presented in terms of a subset S of generators, and we write G = 〈S〉, if every element of G can be expressed as a product of (positive or negative) powers of elements of S. The generators ...
... normal subgroups of G. A group G may be presented in terms of a subset S of generators, and we write G = 〈S〉, if every element of G can be expressed as a product of (positive or negative) powers of elements of S. The generators ...
第 7 頁
... normal subgroup of U and Img T is a subgroup of V. The mapping T : U → V is a monomorphism or “one-to-one” transformation if Ker T contains only the identity element of U; it is an epimorphism or “onto” transformation if Img T is the ...
... normal subgroup of U and Img T is a subgroup of V. The mapping T : U → V is a monomorphism or “one-to-one” transformation if Ker T contains only the identity element of U; it is an epimorphism or “onto” transformation if Img T is the ...
第 18 頁
... μ > 0 has |XY | = μ|XY| for every pair of points. The isometry group of En is the Euclidean group En, a normal subgroup of the Euclidean similarity, or euclopetic, group E ̃n (i.e., if R ∈ En 18 Homogeneous Spaces Isometries.
... μ > 0 has |XY | = μ|XY| for every pair of points. The isometry group of En is the Euclidean group En, a normal subgroup of the Euclidean similarity, or euclopetic, group E ̃n (i.e., if R ∈ En 18 Homogeneous Spaces Isometries.
第 19 頁
... normal subgroup of En. Moreover, as the additive group of the vector space Rn, Tn is abelian; i.e., the translations form a commutative subsystem of isometries. Likewise, a dilatation is a similarity that preserves or reverses ...
... normal subgroup of En. Moreover, as the additive group of the vector space Rn, Tn is abelian; i.e., the translations form a commutative subsystem of isometries. Likewise, a dilatation is a similarity that preserves or reverses ...
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