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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, a Schwarz–Christoffel mapping is a
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
of the
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
or the complex
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...
onto the interior of a
simple polygon In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...
. Such a map is guaranteed to exist by the
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic ma ...
(stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by
Elwin Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provi ...
in 1867 and Hermann Schwarz in 1869. Schwarz–Christoffel mappings are used in
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
and some of its applications, including
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s, hyperbolic art, and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
.


Definition

Consider a polygon in the complex plane. The
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic ma ...
implies that there is a biholomorphic mapping ''f'' from the upper half-plane : \ to the interior of the polygon. The function ''f'' maps the real axis to the edges of the polygon. If the polygon has interior
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s \alpha,\beta,\gamma, \ldots, then this mapping is given by : f(\zeta) = \int^\zeta \frac \,\mathrmw where K is a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
, and a < b < c < \cdots are the values, along the real axis of the \zeta plane, of points corresponding to the vertices of the polygon in the z plane. A transformation of this form is called a ''Schwarz–Christoffel mapping''. The integral can be simplified by mapping the
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. A ...
of the \zeta plane to one of the vertices of the z plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant K. Conventionally, the point at infinity would be mapped to the vertex with angle \alpha. In practice, to find a mapping to a specific polygon one needs to find the a < b < c < \cdots 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
.


Example

Consider a semi-infinite strip in the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes'' ...
. This may be regarded as a limiting form of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
with vertices , , and (with real), as tends to infinity. Now and in the limit. Suppose we are looking for the mapping with , , and . Then is given by : f(\zeta) = \int^\zeta \frac \,\mathrmw. \, Evaluation of this integral yields : z = f(\zeta) = C + K \operatorname \zeta, where is a (complex) constant of integration. Requiring that and gives and . Hence the Schwarz–Christoffel mapping is given by : z = \operatorname \zeta. This transformation is sketched below.


Other simple mappings


Triangle

A mapping to a plane
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
with interior angles \pi a,\, \pi b and \pi(1-a-b) is given by :z=f(\zeta)=\int^\zeta \frac, which can be expressed in terms of hypergeometric functions.


Square

The upper half-plane is mapped to the square by :z=f(\zeta) = \int^\zeta \frac =\sqrt \, F\left(\sqrt;\sqrt/2\right), where ''F'' is the incomplete
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
of the first kind.


General triangle

The upper half-plane is mapped to a triangle with circular arcs for edges by the
Schwarz triangle map In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessar ...
.


See also

* The
Schwarzian derivative In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms a ...
appears in the theory of Schwarz–Christoffel mappings.


References

* * * *
§§267–270, pp. 665–677
*
The Conformal Hyperbolic Square and Its Ilk
Chamberlain Fong, Bridges Finland Conference Proceedings, 2016


Further reading

An analogue of SC mapping that works also for multiply-connected is presented in: .


External links

*
Schwarz–Christoffel toolbox
(software for
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
) {{DEFAULTSORT:Schwarz-Christoffel mapping Conformal mappings