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 34 筆
第 xiv 頁
... rings of real, complex, or quaternionic integers. The associated modular groups are discussed in the final two chapters, which have for the most part been extracted from work done jointly with Asia Ivi ́c Weiss. One of the advantages of ...
... rings of real, complex, or quaternionic integers. The associated modular groups are discussed in the final two chapters, which have for the most part been extracted from work done jointly with Asia Ivi ́c Weiss. One of the advantages of ...
第 5 頁
... matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates and matrix entries may be real numbers, or they may belong to more general number systems, e.g., rings. If Preliminaries 5.
... matrix, and groups of transformations correspond to multiplicative groups of matrices. Coordinates and matrix entries may be real numbers, or they may belong to more general number systems, e.g., rings. If Preliminaries 5.
第 6 頁
... ring 0 has only one element. A ring in which multiplication is commutative is a commutative ring. An integral domain is a nontrivial commutative ring with unity without zero divisors, i.e., such that ab = 0 implies that either a = 0 or ...
... ring 0 has only one element. A ring in which multiplication is commutative is a commutative ring. An integral domain is a nontrivial commutative ring with unity without zero divisors, i.e., such that ab = 0 implies that either a = 0 or ...
第 7 頁
Norman W. Johnson. of a ring homomorphism). The image (or “range”) Img T is the set of elements in V to which elements of U are mapped. When the systems are groups, Ker T is a normal subgroup of U and Img T is a subgroup of V. The ...
Norman W. Johnson. of a ring homomorphism). The image (or “range”) Img T is the set of elements in V to which elements of U are mapped. When the systems are groups, Ker T is a normal subgroup of U and Img T is a subgroup of V. The ...
第 9 頁
... ring. An algebra A is a module over a ring R in which there is also defined a multiplication of module elements, distributive over addition and such that λ(xy) = (λx)y= x(λy) for all λ in R and all x and y in A. If each nonzero element ...
... ring. An algebra A is a module over a ring R in which there is also defined a multiplication of module elements, distributive over addition and such that λ(xy) = (λx)y= x(λy) for all λ in R and all x and y in A. If each nonzero element ...
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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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