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The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of admits a bijective conformal map to the open unit disk in . Informally, this means that any blob can be transformed into a perfect circle by some conformal map.
A Least squares conformal map (LSCM) is a 2-D representation of a 3-D shape created using the Least Squares Conformal Mapping Method. [circular definition] By using the map as a guide when creating a new 2-D image, the colors of the 2-D image can be applied to the original 3-D model.
In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, [1] is a rigidity theorem about conformal mappings in Euclidean space.It states that every smooth conformal mapping on a domain of R n, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions).
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection; that is, the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map ...
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This mapping is known as a Riemann mapping. [1] Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly ...
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves oriented angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian ...