Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.

More formally, a map

$w = f(z),$

is called conformal (or angle-preserving) at $z_0$ if it preserves oriented angles between curves through $z_0$ , 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.

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 manifold.

Complex analysis

An important family of examples of conformal maps comes from complex analysis. If U is an open subset of the complex plane,

mathbb{C}

, then a function

f: U rightarrow mathbb{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.

The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of $mathbb{C}$ admits a bijective conformal map to the open unit disk in $mathbb{C}$ .

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.

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.

Riemannian geometry

In Riemannian geometry, two Riemannian metrics

g

and

h

on smooth manifold

M

are called conformally equivalent if

g = u h

for some positive function

u

M

. The function

u

is called the conformal factor.

A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one.

One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.

For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map.

Higher-dimensional Euclidean space

Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation. (A "special conformal transformation" is the composition of a reflection and an inversion in a sphere.) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

Uses

If a function is harmonic (that is, it satisfies Laplace's equation

nabla^2 f=0

) over a particular space, and is transformed via a conformal map to another space, the transformation is also harmonic. For this reason, any function which is defined by a potential can be transformed by a conformal map and still remain governed by a potential. Examples in physics of equations defined by a potential include the electromagnetic field, the gravitational field, and, in fluid dynamics, potential flow, which is an approximation to fluid flow assuming constant density, zero viscosity, and irrotational flow. One example of a fluid dynamic application of a conformal map is the Joukowsky transform.

The importance of conformal transformations for electromagnetism was brought to light by Harry Bateman in 1910.

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. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example, one may be desirous of calculating the electric field, $E(z),$ arising from a point charge located near the corner of two conducting planes separated by a certain angle (where $z$ is the complex coordinate of a point in 2-space). 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. 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. The solution is obtained in this domain, $E(w),$ and then mapped back to the original domain by noting that $w$ was obtained as a function (viz., the composition of $E$ and $w$ ) of $z,$ whence $E(w)$ can be viewed as $E(w(z)),$ which is a function of $z,$ the original coordinate basis. 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.

In cartography, several named map projections (including the Mercator projection) are conformal.

Alternative angles

A “conformal map” is called that because it conforms to the principle of angle-preservation. The presumption often is that the angle being preserved is the standard Euclidean angle, say parameterized in degrees or radians. However, in plane mapping there are two other angles to consider: the hyperbolic angle and the Galilean angle, which is familiar as the slope used in elementary analytic geometry. Suppose

f: U rightarrow mathbb{V}

is a mapping of surfaces parameterized by

(x,y),

and

(u,v),

. The Jacobian matrix of

f

is formed by the four partial derivatives of

u

and

v

with respect to

x

and

y

. If the Jacobian has a non-zero determinant, then f is “conformal in the generalized sense” with respect to one of the three angle types, depending on the real matrix expressed by the Jacobian g. Indeed, any such g lies in a particular planar commutative subring, and g has a polar coordinate form, g being determined by parameters of radial and angular nature. The radial parameter corresponds to a similarity mapping and can be taken as 1 for purposes of conformal examination. The angular parameter of g is one of the three types, Galilean, hyperbolic, or Euclidean:

When the subring is isomorphic to the dual number plane, then g acts as a shear mapping and preserves the Galilean angle.
When the subring is isomorphic to the split-complex number plane, then g acts as a squeeze mapping and preserves the hyperbolic angle.
When the subring is isomorphic to the ordinary complex number plane, then g acts as a rotation and preserves the Euclidean angle.