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 92 筆
第 2 頁
... given line there can be drawn just one line that does not intersect it. (The other postulates imply the existence of at least one such line.) From these assumptions it follows that the sum of the interior angles of every triangle is ...
... given line there can be drawn just one line that does not intersect it. (The other postulates imply the existence of at least one such line.) From these assumptions it follows that the sum of the interior angles of every triangle is ...
第 3 頁
... given line there is more than one line not intersecting it—we obtain the hyperbolic plane of Bolyai and Lobachevsky. Moreover, if we do not assume that lines are of infinite length, we can construct a metrical geometry in which there ...
... given line there is more than one line not intersecting it—we obtain the hyperbolic plane of Bolyai and Lobachevsky. Moreover, if we do not assume that lines are of infinite length, we can construct a metrical geometry in which there ...
第 5 頁
... given set S form the symmetric group Sym(S), any subgroup of which is a transformation group acting on S. A permutation of a finite set is even or odd according as it can be expressed as the product of an even or an odd number of ...
... given set S form the symmetric group Sym(S), any subgroup of which is a transformation group acting on S. A permutation of a finite set is even or odd according as it can be expressed as the product of an even or an odd number of ...
第 8 頁
... Given an ordered set [x1,...,xk] of vectors in a vector space V, for any ordered set of scalars (λ1 ,..., λk) the vector λ1x1 +···+ λkxk is a linear combination of the vectors. If S is a subset of V and if every vector x in V can be ...
... Given an ordered set [x1,...,xk] of vectors in a vector space V, for any ordered set of scalars (λ1 ,..., λk) the vector λ1x1 +···+ λkxk is a linear combination of the vectors. If S is a subset of V and if every vector x in V can be ...
第 10 頁
... Given two nonzero integers a and b, we can repeatedly apply the division algorithm to obtain a decreasing sequence of positive integers r1 > r2 > ··· > rk, where a = bq1 + r1, b = r1q2 + r2, r1 = r2q3 + r3, ..., r k−1 = rkqk+1 + 0 ...
... Given two nonzero integers a and b, we can repeatedly apply the division algorithm to obtain a decreasing sequence of positive integers r1 > r2 > ··· > rk, where a = bq1 + r1, b = r1q2 + r2, r1 = r2q3 + r3, ..., r k−1 = rkqk+1 + 0 ...
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