Schwarz-Christoffel algorithm

Hydraulics

The Schwarz-Christoffel algorithm is a mathematical method used to map a polygonal region onto a standard geometric shape, such as a disk or a half-plane, in a complex plane. This mapping is conformal, meaning that it preserves angles and shapes locally.

The algorithm takes as input the vertices of the polygonal region, and a set of parameters that determine the shape of the output region. It then computes a function that maps points inside the polygonal region to points inside the output region. This function is given by the Schwarz-Christoffel formula:

f(z) = A ∫ (z — z_0)^(-1/m) * (z — z_1)^(-1/m) * … * (z — z_n)^(-1/m) dz + B

where A and B are complex constants, z_0, z_1, …, z_n are the vertices of the polygonal region, and m is a parameter that determines the shape of the output region.

The formula involves an integral, which can be difficult to evaluate analytically. However, there are numerical methods that can be used to approximate the integral to high accuracy. Once the function f(z) is computed, it can map points inside the polygonal region to points inside the output region.

The Schwarz-Christoffel algorithm has applications in many areas of mathematics and physics, including complex analysis, potential theory, and fluid dynamics. It is also used in computer graphics and image processing, which can map arbitrary shapes onto simpler geometric shapes for easier manipulation and processing.

Coefficient of contraction