Conformal map

In mathematics, a conformal map is a function that preserves orientation and angles locally. In the most common case, the function has a domain and an image in the complex plane. In mathematics, a conformal map is a function that preserves orientation and angles locally. In the most common case, the function has a domain and an image in the complex plane. More formally, let U {displaystyle U} and V {displaystyle V} be open subsets of R n {displaystyle mathbb {R} ^{n}} . A function f : U → V {displaystyle f:U o V} is called conformal (or angle-preserving) at a point u 0 ∈ U {displaystyle u_{0}in U} if it preserves angles between directed curves through u 0 {displaystyle u_{0}} , as well as preserving orientation (i.e. mapping a tangent basis to a basis of the same orientation). 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 an orientation-preserving rotation matrix, then the transformation is conformal. The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds. An important family of examples of conformal maps comes from complex analysis. If U {displaystyle U} is an open subset of the complex plane C {displaystyle mathbb {C} } , then a function f : U → C {displaystyle f:U o mathbb {C} } is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U {displaystyle U} . If f {displaystyle 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 {displaystyle 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 {displaystyle 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 nonzero derivative, but is not one-to-one since it is periodic. The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of C {displaystyle mathbb {C} } admits a bijective conformal map to the open unit disk in C {displaystyle mathbb {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.