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Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Tranformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations'. Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing an elliptic or hyperbolic polarity which transforms points into lines (in two dimensions) or planes (in three dimensions) and vice versa.

Geometry, Non-Euclidean. --- Non-Euclidean geometry --- Geometry --- Parallels (Geometry) --- Foundations

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The epoch-making work of János Bolyai is presented here, together with a supplement outlining Hungarian political and science history to help the reader to get acquainted with the miserable fate of János Bolyai and with his intellectual world. A facsimile of a copy of Bolyai's original 1831 Scientia Spatii (also known as the Appendix) is included, together with a translation. Comments and notes, and a survey of the effects of his work, complete the volume.

Geometry --- Geometry, Non-Euclidean. --- Geometry. --- Mathematics --- Euclid's Elements --- Non-Euclidean geometry --- Parallels (Geometry) --- Foundations

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Starting off from noneuclidean geometries, apart from the method of Einstein's equations, this book derives and describes the phenomena of gravitation and diffraction. A historical account is presented, exposing the missing link in Einstein's construction of the theory of general relativity: the uniformly rotating disc, together with his failure to realize, that the Beltrami metric of hyperbolic geometry with constant curvature describes exactly the uniform acceleration observed. This book also explores these questions: How does time bend? Why should gravity propagate at the speed of light? Ho

Geometry, Non-Euclidean. --- Relativity (Physics) --- Gravitation --- Nonrelativistic quantum mechanics --- Space and time --- Non-Euclidean geometry --- Geometry --- Parallels (Geometry) --- Foundations

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Geometry, Non-Euclidean --- Geometry --- 514.1 --- #KVIV --- Non-Euclidean geometry --- Parallels (Geometry) --- Mathematics --- Euclid's Elements --- General geometry --- Foundations --- 514.1 General geometry

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Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.

Geometry, Hyperbolic --- Géométrie hyperbolique --- Geometry, Hyperbolic. --- Geometry, Non-Euclidean. --- Non-Euclidean geometry --- Geometry --- Parallels (Geometry) --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- Foundations