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 85 筆
第 14 頁
... Figure 1.1a), while two lines in the elliptic plane eP2 intersect in a single point; in each case, the great circles or lines have a unique common ... Figure 1.1b Parallel lines in E2 Figure 1.1d Diverging lines 14 Homogeneous Spaces.
... Figure 1.1a), while two lines in the elliptic plane eP2 intersect in a single point; in each case, the great circles or lines have a unique common ... Figure 1.1b Parallel lines in E2 Figure 1.1d Diverging lines 14 Homogeneous Spaces.
第 15 頁
... becomes Euclidean in the limit (Stäckel 1901; cf. Bonola 1906, §30; Sommerville 1914, p. 15;. Figure 1.1a Great circles in S2 Figure 1.1c Parallel lines in H2 Figure 1.1e Ordinary, absolute, and ideal lines. 1.1 Real Metric Spaces 15.
... becomes Euclidean in the limit (Stäckel 1901; cf. Bonola 1906, §30; Sommerville 1914, p. 15;. Figure 1.1a Great circles in S2 Figure 1.1c Parallel lines in H2 Figure 1.1e Ordinary, absolute, and ideal lines. 1.1 Real Metric Spaces 15.
第 17 頁
... Figure 1.1e. The absolute hypersphere is the locus of self-conjugate points in an absolute polarity that pairs each ordinary, absolute, or ideal point with an ideal, absolute, or ordinary hyperplane. Indeed, as shown by Cayley (1859) ...
... Figure 1.1e. The absolute hypersphere is the locus of self-conjugate points in an absolute polarity that pairs each ordinary, absolute, or ideal point with an ideal, absolute, or ordinary hyperplane. Indeed, as shown by Cayley (1859) ...
第 29 頁
... figures. A complete quadrangle PQRS is a set of four points P, Q, R, S, no three collinear, and the six lines QR and ... Figure 2.1a), let S be any point not lying on S,ˇ and let R be any point on the line Cˇ = CS other than C or S. Let ...
... figures. A complete quadrangle PQRS is a set of four points P, Q, R, S, no three collinear, and the six lines QR and ... Figure 2.1a), let S be any point not lying on S,ˇ and let R be any point on the line Cˇ = CS other than C or S. Let ...
第 30 頁
... choice of the auxiliary points S and R in the above construction. An affirmative answer requires still another assumption: Axiom M. Each point of a harmonic set is uniquely. Figure 2.1a The harmonic construction. 30 Linear Geometries.
... choice of the auxiliary points S and R in the above construction. An affirmative answer requires still another assumption: Axiom M. Each point of a harmonic set is uniquely. Figure 2.1a The harmonic construction. 30 Linear Geometries.
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