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 60 筆
第 vii 頁
... .................. 65 3.4 Triangles and Trigonometry ...................... 74 3.5 Non-Euclidean Circles ......................... 81 3.6 Summary of Real Spaces ........................ 86 4 Real Collineation Groups .........
... .................. 65 3.4 Triangles and Trigonometry ...................... 74 3.5 Non-Euclidean Circles ......................... 81 3.6 Summary of Real Spaces ........................ 86 4 Real Collineation Groups .........
第 1 頁
... triangles, circles, and spheres. Among the fundamental concepts of Euclidean plane geometry are collinearity, congruence, perpendicularity, and parallelism. A rigorous treatment also involves order and continuity—relations not ...
... triangles, circles, and spheres. Among the fundamental concepts of Euclidean plane geometry are collinearity, congruence, perpendicularity, and parallelism. A rigorous treatment also involves order and continuity—relations not ...
第 2 頁
... triangles are equivalent, as are all parallelograms; there is no such thing as a right triangle or a square. Conics can be distinguished only as ellipses, parabolas, and hyperbolas—there are no circles. By adopting the convention that ...
... triangles are equivalent, as are all parallelograms; there is no such thing as a right triangle or a square. Conics can be distinguished only as ellipses, parabolas, and hyperbolas—there are no circles. By adopting the convention that ...
第 3 頁
... triangle depends on its area, being proportionally greater than two right angles for an elliptic (spherical) triangle or proportionally less for a hyperbolic one. Other properties of Euclidean and non-Euclidean geometries (“real metric ...
... triangle depends on its area, being proportionally greater than two right angles for an elliptic (spherical) triangle or proportionally less for a hyperbolic one. Other properties of Euclidean and non-Euclidean geometries (“real metric ...
第 28 頁
... triangle PQR or PˇQˇRˇ is a set of three noncollinear points P, Q, R and the three lines Pˇ= QR, Qˇ = RP, Rˇ = PQ joining them in pairs or, dually, a set of three nonconcurrent lines P,ˇ Q,ˇ Rˇ and the three points P = Q·ˇR,ˇ Q = R·ˇP,ˇ ...
... triangle PQR or PˇQˇRˇ is a set of three noncollinear points P, Q, R and the three lines Pˇ= QR, Qˇ = RP, Rˇ = PQ joining them in pairs or, dually, a set of three nonconcurrent lines P,ˇ Q,ˇ Rˇ and the three points P = Q·ˇR,ˇ Q = R·ˇP,ˇ ...
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