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 21 筆
第 xi 頁
... axioms for the real projective plane. In developing the general theory of projective n-space and other spaces, however, I make free use of coordinates, mappings, and other algebraic concepts, including cross ratios. Like Euclidean space ...
... axioms for the real projective plane. In developing the general theory of projective n-space and other spaces, however, I make free use of coordinates, mappings, and other algebraic concepts, including cross ratios. Like Euclidean space ...
第 27 頁
... 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 or affine n-space into a linear metric space—elliptic, hyperbolic, or Euclidean n ...
... 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 or affine n-space into a linear metric space—elliptic, hyperbolic, or Euclidean n ...
第 28 頁
... Axiom I. Any two distinct points are incident with just one line. Axiom J. Any two lines are incident with at least one point. Axiom K. There exist four points no three of which are collinear. Definitions, descriptions, and statements ...
... Axiom I. Any two distinct points are incident with just one line. Axiom J. Any two lines are incident with at least one point. Axiom K. There exist four points no three of which are collinear. Definitions, descriptions, and statements ...
第 30 頁
... Axiom”: Axiom L. The three diagonal points of a complete quadrangle are never collinear. From Axioms I, J, K, and L we can easily prove the dual of Axiom L: Theorem L.ˇ The three diagonal lines of a complete quadrilateral are never ...
... Axiom”: Axiom L. The three diagonal points of a complete quadrangle are never collinear. From Axioms I, J, K, and L we can easily prove the dual of Axiom L: Theorem L.ˇ The three diagonal lines of a complete quadrilateral are never ...
第 31 頁
... Axiom M is Axiom N. Each point of a quadrangular set is uniquely determined by the remaining points. Since a harmonic set is a special case of a quadrangular set, Axiom N implies Axiom M. Given Axioms I, J, K, and N (but without ...
... Axiom M is Axiom N. Each point of a quadrangular set is uniquely determined by the remaining points. Since a harmonic set is a special case of a quadrangular set, Axiom N implies Axiom M. Given Axioms I, J, K, and N (but without ...
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