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Equiareal Map
In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures. Properties If ''M'' and ''N'' are two Riemannian (or pseudo-Riemannian) surfaces, then an equiareal map ''f'' from ''M'' to ''N'' can be characterized by any of the following equivalent conditions: * The surface area of ''f''(''U'') is equal to the area of ''U'' for every open set ''U'' on ''M''. * The pullback of the area element ''μ''''N'' on ''N'' is equal to ''μ''''M'', the area element on ''M''. * At each point ''p'' of ''M'', and tangent vectors ''v'' and ''w'' to ''M'' at ''p'',\bigl, df_p(v)\wedge df_p(w)\bigr, = , v\wedge w, \,where \wedge denotes the Euclidean wedge product of vectors and ''df'' denotes the pushforward along ''f''. Example An example of an equiareal map, due to Archimedes of Syracuse, is the projection from the unit sphere to the unit cylinder outward from their common axis. An explicit for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
