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 1 到 5 筆結果,共 35 筆
第 xii 頁
... expression in convenient notations for coordinates and for corresponding bilinear and quadratic forms. The pseudo-symmetry of real and imaginary is likewise evident in inversive geometry as well as in the metric properties of hyperbolic ...
... expression in convenient notations for coordinates and for corresponding bilinear and quadratic forms. The pseudo-symmetry of real and imaginary is likewise evident in inversive geometry as well as in the metric properties of hyperbolic ...
第 xiii 頁
... expressed as the product of reffections, and discrete groups generated by reffections— known as Coxeter groups—are of particular importance. Finite and infinite symmetry groups are subgroups or extensions of spherical and Euclidean ...
... expressed as the product of reffections, and discrete groups generated by reffections— known as Coxeter groups—are of particular importance. Finite and infinite symmetry groups are subgroups or extensions of spherical and Euclidean ...
第 5 頁
... expressed as a product of (positive or negative) powers of elements of S. The generators satisfy certain relations, and G is the largest group for which the specified relations (but no others independent of them) hold. Thus the cyclic ...
... expressed as a product of (positive or negative) powers of elements of S. The generators satisfy certain relations, and G is the largest group for which the specified relations (but no others independent of them) hold. Thus the cyclic ...
第 7 頁
... expressed as algebraic homomorphisms, successive operations are normally carried out from left to right, as in the diagram U 1−→V T 2−→W T The product (T1 followed by T2) of the homomorphisms T 1 : U → V and T2 : V → W is then the ...
... expressed as algebraic homomorphisms, successive operations are normally carried out from left to right, as in the diagram U 1−→V T 2−→W T The product (T1 followed by T2) of the homomorphisms T 1 : U → V and T2 : V → W is then the ...
第 8 頁
... expressed as a linear combination of vectors in S, the set S spans V. If a linear combination of distinct vectors xi in a set S is the zero vector only when all the scalar coefficients λi are zero, the vectors in S are linearly ...
... expressed as a linear combination of vectors in S, the set S spans V. If a linear combination of distinct vectors xi in a set S is the zero vector only when all the scalar coefficients λi are zero, the vectors in S are linearly ...
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