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 93 筆
第 14 頁
... given line, the limit of a circle through a fixed point in E2 is a straight line, but in H2 the limit is a horocycle— a “circle” with its center at infinity. Similarly, the locus of points at a constant distance from a given line in E2 ...
... given line, the limit of a circle through a fixed point in E2 is a straight line, but in H2 the limit is a horocycle— a “circle” with its center at infinity. Similarly, the locus of points at a constant distance from a given line in E2 ...
第 19 頁
... given point fixed. The center of On—the subgroup of isometries of Sn−1 that commute with every isometry—is the unit scalar (or “central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by ...
... given point fixed. The center of On—the subgroup of isometries of Sn−1 that commute with every isometry—is the unit scalar (or “central”) group SZ, ̄ the group of order 2 generated by the central inversion, sometimes denoted simply by ...
第 20 頁
Norman W. Johnson. The set of images of a given point of a metric space under the operations of a group of isometries is its orbit. If every point has a neighborhood that includes no other points in its orbit, the group is said to be ...
Norman W. Johnson. The set of images of a given point of a metric space under the operations of a group of isometries is its orbit. If every point has a neighborhood that includes no other points in its orbit, the group is said to be ...
第 21 頁
... given line can serve as the axis of the translation. In Hn (n ≥ 2), the mirrors are diverging, and the axis is unique. Except for the identity, a translation is of infinite (or imaginary) period. In the Euclidean case, repeated ...
... given line can serve as the axis of the translation. In Hn (n ≥ 2), the mirrors are diverging, and the axis is unique. Except for the identity, a translation is of infinite (or imaginary) period. In the Euclidean case, repeated ...
第 28 頁
... given line, the line is said to join the points, while two lines incident with a given point are said to meet (or “intersect”) in the point. Three or more points lying on one line are collinear, and three or more lines passing through ...
... given line, the line is said to join the points, while two lines incident with a given point are said to meet (or “intersect”) in the point. Three or more points lying on one line are collinear, and three or more lines passing through ...
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