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 36 筆
第 4 頁
... operation G × G → G, with (a, b) ↦→ ab, satisfying the associative law (ab)c = a(bc), having an identity element, and with each element having a unique inverse. The group operation is commonly taken as multiplication, with the ...
... operation G × G → G, with (a, b) ↦→ ab, satisfying the associative law (ab)c = a(bc), having an identity element, and with each element having a unique inverse. The group operation is commonly taken as multiplication, with the ...
第 6 頁
... operation but possibly lacking an identity element or inverses, then a ring R is a set whose elements form both an additive abelian group and a multiplicative semigroup, satisfying the distributive laws a(b+c) = ab+ ac and (a + b)c = ac ...
... operation but possibly lacking an identity element or inverses, then a ring R is a set whose elements form both an additive abelian group and a multiplicative semigroup, satisfying the distributive laws a(b+c) = ab+ ac and (a + b)c = ac ...
第 7 頁
... 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 homomorphism T 1 T2 : U → W, with x(T1 T2) ...
... 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 homomorphism T 1 T2 : U → W, with x(T1 T2) ...
第 8 頁
... operations—expressed algebraically as linear transformations of coordinates—are represented by matrices. Basic geometric properties, such as distances and angles, are defined by means of bilinear forms, functions V × V → F that map ...
... operations—expressed algebraically as linear transformations of coordinates—are represented by matrices. Basic geometric properties, such as distances and angles, are defined by means of bilinear forms, functions V × V → F that map ...
第 19 頁
... a suitable stem, the suffix “-petic” (from the Greek anapeteia expansion) indicates the result of combining a measure-preserving transformation and a scaling operation. The set of images of a given point of a 1.2 Isometries 19.
... a suitable stem, the suffix “-petic” (from the Greek anapeteia expansion) indicates the result of combining a measure-preserving transformation and a scaling operation. The set of images of a given point of a 1.2 Isometries 19.
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