A Treatise on the Circle and the Sphere (AMS Chelsea Publishing 236)
By: Julian Lowell Coolidge (author)Hardback
1 - 2 weeks availability
Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's sphere geometry, whose group of transformations is precisely the conformal group. Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still widely felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles orthogonal to a given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry, and Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.
The Circle in Elementary Plane Geometry: Fundamental definitions and notation Inversion Mutually tangent circles Circles related to a triangle The Brocard figures Concurrent circles and concyclic points Coaxal circles The Circle in Cartesian Plane Geometry: The circle studied by means of trilinear coordinates Fundamental relations, special tetracyclic coordinates The identity of Darboux and Frobenius Analytic systems of circles Famous Problems in Construction: Lemoine's geometrographic criteria Problem of Apollonius, number of real solutions Construction of Apollonius Construction of Gergonne Steiner's problem Circle meeting four others at equal or supplementary angles Malfatti's problem, Hart's proof of Steiner's construction Analytic solution, extension to thirty-two cases Examples of Fiedler's general cyclographic methods Mascheroni's geometry of the compass The Tetracyclic Plane: Fundamental theorems and definitions Cyclics The Sphere in Elementary Geometry: Miscellaneous elementary theorems Coaxal systems The Sphere in Cartesian Geometry: Coordinate systems Identity of Darboux and Frobenius Analytic systems of spheres Pentaspherical Space: Fundamental definitions and theorems Cyclides Circle Transformations: General theory Analytic treatment Continuous groups of transformations Sphere Transformations: General theory Continuous groups The Oriented Circle: Elementary geometrical theory Analytic treatment Laguerre transformations Continuous groups Hypercyclics The oriented circle treated directly The Oriented Sphere: Elementary geometrical theorems Analytic treatment The hypercyclide The oriented sphere treated directly Line-sphere transformation Complexes of oriented spheres Circles Orthogonal to One Sphere: Relations of two circles Circles orthogonal to one sphere Systems of circle crosses Circles in Space, Algebraic Systems: Coordinates and identities Linear systems of circles Other simple systems The Oriented Circle in Space: Fundamental relations Linear systems The Laguerre method of representing imaginary points Differential Geometry of Circle Systems: Differential geometry of thecircles Parametric method for circle congruences The Kummer method Complexes of circles Subject index Index of proper names.
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