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This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. This title, first published in 2016, has been reissued as an Open Access publication on Cambridge Core.

Conformal geometry. --- Conformal mapping. --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry

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The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

Conformal mapping. --- Quaternions. --- 514.1 --- Conformal mapping --- Quaternions --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- 514.1 General geometry --- General geometry --- Differential geometry. --- Differential Geometry. --- Differential geometry

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In the context of Geographical Information Systems (GIS) the book offers a timely review of Map Projections. The first chapters are of foundational type. We introduce the mapping from a left Riemann manifold to a right one specified as conformal, equiaerial and equidistant, perspective and geodetic. In particular, the mapping from a Riemann manifold to a Euclidean manifold ("plane") and the design of various coordinate systems are reviewed . A speciality is the treatment of surfaces of Gaussian curvature zero. The largest part is devoted to the mapping the sphere and the ellipsoid-of-revolution to tangential plane, cylinder and cone (pseudo-cone) using the polar aspect, transverse as well as oblique aspect. Various Geodetic Mappings as well as the Datum Problem are reviewed. In the first extension we introduce optimal map projections by variational calculus for the sphere, respectively the ellipsoid generating harmonic maps. The second extension reviews alternative maps for structures ,  namely torus (pneu), hyperboloid (cooling tower), paraboloid (parabolic mirror), onion shape (church tower) as well as clothoid (Hight Speed Railways) used in Project Surveying. Third, we present the Datum Transformation described by the Conformal Group C10 (3) in a threedimensional Euclidean space , a ten parameter conformal transformation. It leaves infinitesimal angles and distance ratios equivariant. Numerical examples from classical and new map projections as well as twelve appendices document the Wonderful World of Map Projections.

Map projection --- Cartography --- Conformal mapping --- Data processing. --- Mathematics. --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Cartography, Primitive --- Chartography --- Map-making --- Mapmaking --- Mapping (Cartography) --- Mathematical geography --- Surveying --- Maps --- Projection --- Geographical information systems. --- Physical geography. --- Geography. --- Geographical Information Systems/Cartography. --- Geophysics/Geodesy. --- Geography, general. --- Cosmography --- Earth sciences --- World history --- Geography --- Geographical information systems --- GIS (Information systems) --- Information storage and retrieval systems --- Geophysics. --- Geological physics --- Terrestrial physics --- Physics

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Complex analysis --- Kernel functions --- #TCPW W5.0 --- 517.54 --- Functions, Kernel --- Functions of complex variables --- Geometric function theory --- Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Conformal mapping. --- Kernel functions. --- 517.54 Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Conformal mapping --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Surfaces, Representation of --- Mathematical logic --- Mappings (Mathematics) --- Transformations (Mathematics) --- 517.5 --- 517.5 Theory of functions --- Theory of functions --- Noyaux (analyse fonctionnelle) --- Fonctions d'une variable complexe --- Representation conforme

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Functions of complex variables --- Potential theory (Mathematics) --- Cauchy transform --- Conformal mapping --- Cauchy transform. --- Conformal mapping. --- Functions of complex variables. --- 517.54 --- Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Potential theory (Mathematics). --- 517.54 Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Complex variables --- Elliptic functions --- Functions of real variables --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Cauchy-Hilbert transform --- Cauchy's transform --- Transform, Cauchy