A rectangular grid (top) and its image under a conformal map (bottom). It is seen that maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
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In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let and be open subsets of . A function is called conformal (or angle-preserving) at a point if it preserves angles between directed curves through , 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.[1]
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.
^Blair, David (2000-08-17). Inversion Theory and Conformal Mapping. The Student Mathematical Library. Vol. 9. Providence, Rhode Island: American Mathematical Society. doi:10.1090/stml/009. ISBN 978-0-8218-2636-2. S2CID 118752074.
orientation. Conformalmaps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property...
is a conformalmap in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map with a conformal projection...
A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and...
needed] The most well-known map projection is the Mercator projection.: 45 This map projection has the property of being conformal. However, it has been criticized...
Look up conformal in Wiktionary, the free dictionary. Conformal may refer to: Conformal (software), in ASIC Software Conformal coating in electronics Conformal...
A Least squares conformalmap (LSCM) is a 2-D representation of a 3-D shape created using the Least Squares Conformal Mapping Method.[circular definition]...
mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes...
Fuller projection Dymaxion-like conformal projection Comparison of the Fuller projection and Strebe's Dymaxion-like conformal projection with Tissot's indicatrices...
transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group. If...
orientation. Conformalmaps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property...
also known as the planisphere projection or the azimuthal conformal projection, is a conformalmap projection whose use dates back to antiquity. Like the...
is a conformal cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection...
regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures. Every...
{\displaystyle f} is biholomorphic implies that it is a conformalmap and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any...
conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry...
transformation may refer to: Bilinear map or bilinear operator Bilinear transform (signal processing), a type of conformalmap used to switch between continuous-time...
The Lee conformal world in a tetrahedron is a polyhedral, conformalmap projection that projects the globe onto a tetrahedron using Dixon elliptic functions...
physics, certain conformalmaps known as spherical wave transformations are special conformal transformations. A special conformal transformation can...
are some of the most important topics in geometric function theory: A conformalmap is a function which preserves angles locally. In the most common case...
\colon \mathbb {C} ^{*}\to S} . The conformal mapping section below explains the geometric properties of this map in more detail. On the region C − R...
sphere, using particular projection to map each face with low distortion. Conformal Preserves angles locally, implying that local shapes are not distorted...
conformalmap as one with nonzero derivative, but without requiring that the map be injective. According to this weaker definition, a conformalmap need...
The Peirce quincuncial projection is the conformalmap projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce...
transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformalmap historically used to understand some principles of airfoil design. It...
and medical imaging. Computational quasi-conformal geometry has been developed, which extends the quasi-conformal theory into a discrete setting. It has...
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