Schwarz–Christoffel mapping

A conformal map from a half-plane or disk to a polygon’s interior. Its derivative is a product of powers whose exponents encode the polygon’s interior angles.

Schwarz–Christoffel formula: An integral formula mapping the upper half-plane to a polygon, with derivative factors determined by its vertex angles. Its boundary factors produce the polygon’s prescribed turning angles.
Complex analysis: The study of functions of complex variables, including their differentiability, integrals, and singularities. The mapping formula is built from analytic functions and complex integration.
Laplace's equation: The partial differential equation stating that a function’s Laplacian is zero. Conformal maps transfer two-dimensional harmonic boundary-value problems to polygonal domains.
Joukowsky transform: The complex map z ↦ z + 1/z, used in models of airfoils and related shapes. It maps specific geometries by an elementary formula rather than encoding arbitrary polygon angles.
Prevertices: Boundary points in the source domain that map to polygon vertices under a Schwarz–Christoffel map. Their locations determine the polygon’s shape and create the accessory-parameter problem.
Upper half-plane: The set of complex numbers with positive imaginary part. It is a standard source domain whose real boundary maps to polygon edges.
Potential flow: An idealized fluid flow described by a velocity potential whose gradient gives the velocity field. Polygonal channels can be transformed into simpler flow domains.
Power function: A function of the form z ↦ z^α for a constant exponent α. A single power maps a wedge, whereas the polygon formula combines factors for many vertices.
Interior angle: The angle enclosed by two sides of a polygon at one of its vertices. Each angle fixes an exponent in the mapping derivative.
Unit disk: The set of complex numbers whose modulus is less than one. A Möbius transformation can convert it to the half-plane formulation.

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