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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain and an image in the complex plane. More formally, a map : ''f'' : ''U'' → ''V'' with ''U'', ''V'' ⊆ ℂ''n'' is called conformal (or angle-preserving) at a point ''u''0 if it preserves oriented angles between curves through ''u''0 with respect to their orientation (i.e. not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold. ==Complex analysis== An important family of examples of conformal maps comes from complex analysis. If ''U'' is an open subset of the complex plane ℂ, then a function : ''f'' : ''U'' → ℂ is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on ''U''. If ''f'' is antiholomorphic (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps; a map f defined on an open set is said to be conformal if it is one-to-one and holomorphic. Since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function (defined on the image of f) to be holomorphic. Thus, under this definition, a map is conformal if and only if it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative. However, the exponential function is a holomorphic function with a non-zero derivative, but is not one-to-one since it is periodic. 〔http://www.maths.tcd.ie/~richardt/414/414-ch7.pdf〕 The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of ℂ admits a bijective conformal map to the open unit disk in ℂ. A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation. Again, for the conjugate, angles are preserved, but orientation is reversed. An example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the radial coordinate in circular coordinates, keeping the angle the same. See also inversive geometry. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「conformal map」の詳細全文を読む スポンサード リンク
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