Conformal transformations

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
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\to 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. 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. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types.
The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds.

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