Computational Conformal Mapping
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Computational Conformal Mapping for Surface Grid Generation
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. An important family of examples of conformal maps comes from complex analysis.
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.
Computational Conformal Mapping for Surface Grid Generation
However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. 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.
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A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map. One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.
A classical theorem of Joseph Liouville called Liouville's theorem shows the higher-dimensions have less varied conformal maps:. Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: A special conformal transformation is the composition of a reflection and an inversion in a sphere.
However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely pi radians, meaning that the corner of two planes is transformed to a straight line. The paper describes the development and application of a new approach for formulating an elliptic generation system on parametrically defined surfaces. The class consisted of students from applied mathematics andengineering. As recounted by Andrew Warwick Masters of Theory: Linear Operators, Part 3:
Thus, the set of conformal transformations in spaces of dimension greater than 2 is much more restricted than in the planar case, where the Riemann mapping theorem provides a large set of conformal transformations. In cartography , several named map projections , including the Mercator projection and the stereographic projection are conformal. These enjoy the property that the distortion of shapes can be made as small as desired by making the diameter of the mapped region small enough.
See Conformal map projection. Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries.
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By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. This problem per se is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely pi radians, meaning that the corner of two planes is transformed to a straight line.
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In this new domain, the problem that of calculating the electric field impressed by a point charge located near a conducting wall is quite easy to solve. Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary. Another example is the application of conformal mapping technique for solving the boundary value problem of liquid sloshing in tanks.
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