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In mathematics, a conformal map is a function which preserves angles. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
More formally, a map
- w = f(z)
is called conformal (or angle-preserving) at z0, if it preserves oriented angles between curves through z0, as well as their orientation, i.e. direction. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size. This article is about angles in geometry. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
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. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
Cartography
In cartography, a conformal map projection is a map projection that preserves the angles at all but a finite number of points. The scale depends on location, but not on direction. Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ...
The Mercator projection shows courses of constant bearing as straight lines. ...
This article is about angles in geometry. ...
Examples include the Mercator projection and the stereographic projection. The Mercator projection of the world up to a latitude of 86° N and S The Mercator projection is a cylindrical map projection devised by Gerardus Mercator in 1569. ...
Stereographic projection of a circle of radius R onto the x axis. ...
Complex analysis An important family of examples of conformal maps comes from complex analysis. If U is an open subset of the complex plane, C, then a function This article may be too technical for most readers to understand. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
- f : U → C
is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In mathematics, the derivative is one of the two central concepts of calculus. ...
In mathematics, a function on the complex plane is antiholomorphic at a point if its derivative with respect to z* exists, where here, z* is the complex conjugate. ...
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of C admits a bijective conformal map to the open unit disk in C. The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
A disc of unit radius on a plane is called a unit disc. ...
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. In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
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. This article describes some of the common coordinate systems that appear in elementary mathematics. ...
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ...
Riemannian geometry In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called conformally equivalent if for g=uh for some positive function u on M. The function u is called conformal factor. In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map. Stereographic projection of a circle of radius R onto the x axis. ...
A sphere is a perfectly symmetrical geometrical object. ...
This article is in need of attention from an expert on the subject. ...
The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...
Euclidean space Any conformal map from Euclidean space of dimension at least 3 to itself is a composition of a homothetic transformation and an isometry. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
Uses The importance of conformal transformations for electromagnetism was brought to light by Harry Bateman in 1910. A quadrupole (four-pole) electromagnet, used to focus particle beams in a particle accelerator. ...
Harry Bateman (May 29, 1882 Manchester, England - January 21, 1946 Pasadena California USA) was a leading English mathematician. ...
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