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 6 到 10 筆結果,共 34 筆
第 27 頁
... meet in at most one point. The real projective plane is based on a few elementary axioms, but for higher dimensions we make use of coordinates. In this chapter we shall see how the selection of an absolute polarity converts projective ...
... meet in at most one point. The real projective plane is based on a few elementary axioms, but for higher dimensions we make use of coordinates. In this chapter we shall see how the selection of an absolute polarity converts projective ...
第 28 頁
... meet (or “intersect”) in the point. Three or more points lying on one line are collinear, and three or more lines passing through one point are concurrent. In any projective plane, we make the following assumptions of incidence (cf ...
... meet (or “intersect”) in the point. Three or more points lying on one line are collinear, and three or more lines passing through one point are concurrent. In any projective plane, we make the following assumptions of incidence (cf ...
第 29 頁
... meet are the quadrangle's diagonal points. A complete quadrangle is a configuration (43,62) of four points and six lines, with three lines through each point and two points on each line. Dually, a complete quadrilateral PˇQˇRˇSˇ is a ...
... meet are the quadrangle's diagonal points. A complete quadrangle is a configuration (43,62) of four points and six lines, with three lines through each point and two points on each line. Dually, a complete quadrilateral PˇQˇRˇSˇ is a ...
第 31 頁
... meet are collinear. Given Axioms I, J, K, L, and M, one can prove the following result, a special case of the celebrated two-triangle theorem of Girard Desargues (1593–1662): Theorem 2.1.1 (Minor Theorem of Desargues). If two triangles ...
... meet are collinear. Given Axioms I, J, K, L, and M, one can prove the following result, a special case of the celebrated two-triangle theorem of Girard Desargues (1593–1662): Theorem 2.1.1 (Minor Theorem of Desargues). If two triangles ...
第 32 頁
... meet, and the center and axis of perspectivity form a Desargues configuration 103 of ten points and ten lines, with three lines through each point and three points on each line (see Figure 2.1c). When the center lies on the axis, as in ...
... meet, and the center and axis of perspectivity form a Desargues configuration 103 of ten points and ten lines, with three lines through each point and three points on each line (see Figure 2.1c). When the center lies on the axis, as in ...
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