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 52 筆
第 xiv 頁
... 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 building a theory of geometries and ...
... 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 building a theory of geometries and ...
第 8 頁
... integer n, the canonical vector space F n comprises all lists (x1,...,xn) of n elements of F (the “entries” of the list), with element-by-element addition and scalar multiplication. The vector space F 1 is the field F itself. Given an ...
... integer n, the canonical vector space F n comprises all lists (x1,...,xn) of n elements of F (the “entries” of the list), with element-by-element addition and scalar multiplication. The vector space F 1 is the field F itself. Given an ...
第 10 頁
... integer b is a divisor of an integer a if there is an integer c such that a = bc. A positive integer p greater than 1 whose only positive divisors are 1 and p itself is a prime. If a and b are integers with b > 0, then there exist unique ...
... integer b is a divisor of an integer a if there is an integer c such that a = bc. A positive integer p greater than 1 whose only positive divisors are 1 and p itself is a prime. If a and b are integers with b > 0, then there exist unique ...
第 11 頁
... integer can be uniquely factored into the product of powers of primes. These ideas can be extended to other integral domains and number systems, but not every system allows a version of the Euclidean algorithm or has the unique ...
... integer can be uniquely factored into the product of powers of primes. These ideas can be extended to other integral domains and number systems, but not every system allows a version of the Euclidean algorithm or has the unique ...
第 21 頁
... integer m such that it can be effected in En or Hn (or ePn if of odd type) for any n ≥ m or in Sn (or ePn if of even type) for any n ≥ m−1. Table 1.2 lists the type and rank of every nontrivial isometry of Sn (or ePn), En, or Hn. The ...
... integer m such that it can be effected in En or Hn (or ePn if of odd type) for any n ≥ m or in Sn (or ePn if of even type) for any n ≥ m−1. Table 1.2 lists the type and rank of every nontrivial isometry of Sn (or ePn), En, or Hn. The ...
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