Schwarz–Christoffel mapping
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem ; the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869.
Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics.
Definition
Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-planeto the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles, then this mapping is given by
where is a constant, and are the values, along the real axis of the plane, of points corresponding to the vertices of the polygon in the plane. A transformation of this form is called a Schwarz–Christoffel mapping.
The integral can be simplified by mapping the point at infinity of the plane to one of the vertices of the plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant. Conventionally, the point at infinity would be mapped to the vertex with angle.
In practice, to find a mapping to a specific polygon one needs to find the values which generate the correct polygon side lengths. This requires solving a set of nonlinear equations, and in most cases can only be done numerically.
Example
Consider a semi-infinite strip in the plane. This may be regarded as a limiting form of a triangle with vertices,, and, as tends to infinity. Now and in the limit. Suppose we are looking for the mapping with,, and. Then is given byEvaluation of this integral yields
where is a constant of integration. Requiring that and gives and. Hence the Schwarz–Christoffel mapping is given by
This transformation is sketched below.
Other simple mappings
Triangle
A mapping to a plane triangle with interior angles and is given bywhich can be expressed in terms of hypergeometric functions or incomplete beta functions.
The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.
Square
The upper half-plane is mapped to the square bywhere F is the incomplete elliptic integral of the first kind.