8.3 Schwarz-Christoffel transformation

The Schwarz-Christoffel transformation is a powerful tool for mapping the upper half-plane or unit disk onto simple polygons. It's derived by considering the behavior of the mapping function at polygon vertices, with interior angles determining the formula's exponents.

This transformation is crucial for solving boundary value problems in polygonal domains. By transforming complex geometries into simpler ones, it simplifies problem-solving in electrostatics, fluid dynamics, and heat conduction, making it a key technique in conformal mapping.

Schwarz-Christoffel Formula Derivation

Conformal Mapping and Polygon Vertices

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• The Schwarz-Christoffel formula maps the upper half-plane or unit disk conformally onto the interior of a simple polygon
• The formula is derived by considering the behavior of the mapping function at the vertices of the polygon
• The interior angles of the polygon determine the exponents in the Schwarz-Christoffel formula
• Example: A square with interior angles of 90° (π/2 radians) at each vertex

Derivative and Integration of the Mapping Function

• The derivative of the mapping function is expressed as a product of power functions
• Each factor in the product corresponds to a vertex of the polygon
• The exponents in the formula are related to the interior angles of the polygon by the equation $\alpha_k = 1 - (\theta_k / \pi)$, where $\theta_k$ is the interior angle at the k-th vertex
• The mapping function is obtained by integrating the derivative and introducing appropriate constants to ensure the desired correspondence between the domain and the polygon
• Example: For a triangle with interior angles $\theta_1$, $\theta_2$, and $\theta_3$, the derivative of the mapping function is $f'(z) = C(z-z_1)^{\alpha_1-1}(z-z_2)^{\alpha_2-1}(z-z_3)^{\alpha_3-1}$

Schwarz-Christoffel Parameter Calculation

Prevertices and Multiplicative Constant

• The Schwarz-Christoffel parameters include the prevertices (points in the upper half-plane or unit disk that map to the vertices of the polygon) and the multiplicative constant
• The prevertices are typically chosen to simplify the calculations, such as placing them at convenient points like 0, 1, and ∞ for a triangle
• The multiplicative constant is determined by ensuring that the mapping function takes on the correct values at specific points, such as mapping the prevertices to the corresponding vertices of the polygon
• Example: For a square, the prevertices can be chosen as -1, 0, 1, and ∞, with the multiplicative constant determined by the side length of the square

Solving Nonlinear Equations and Numerical Methods

• For polygons with more than three vertices, the prevertices and multiplicative constant are often determined by solving a system of nonlinear equations
• Numerical methods, such as the Schwarz-Christoffel toolbox in MATLAB, can be used to calculate the parameters for complex polygons
• These methods typically involve iterative algorithms that minimize the error between the desired polygon and the one obtained by the current set of parameters
• Example: The Schwarz-Christoffel toolbox in MATLAB can compute the parameters for polygons with arbitrary number of vertices and interior angles

Mapping with Schwarz-Christoffel Transformation

Applying the Transformation

• The Schwarz-Christoffel transformation is applied by substituting the calculated parameters into the general formula
• For the upper half-plane, the transformation maps the real axis to the boundary of the polygon and the upper half-plane to the interior of the polygon
• For the unit disk, the transformation maps the unit circle to the boundary of the polygon and the interior of the disk to the interior of the polygon
• Example: For a polygon with vertices $w_1, w_2, ..., w_n$ and prevertices $z_1, z_2, ..., z_n$, the Schwarz-Christoffel transformation is given by $f(z) = C \int \prod_{k=1}^n (z-z_k)^{\alpha_k-1} dz$

Properties of the Mapping

• The mapping preserves angles (conformal) and is one-to-one inside the domain, ensuring that each point in the upper half-plane or unit disk corresponds to a unique point in the polygon
• The behavior of the mapping function near the vertices of the polygon is determined by the exponents in the formula
• Smaller interior angles result in a more severe crowding of the preimages near the corresponding prevertex
• Example: In a rectangle, the preimages of the vertices with 90° angles are more evenly distributed compared to those of a thin triangle with small angles

Boundary Value Problems with Schwarz-Christoffel Transformation

Transforming and Solving Boundary Value Problems

• The Schwarz-Christoffel transformation is a powerful tool for solving boundary value problems in polygonal domains
• The problem is first transformed from the polygonal domain to the upper half-plane or unit disk using the Schwarz-Christoffel mapping
• In the transformed domain, the boundary value problem often becomes simpler to solve, as the boundary conditions are now imposed on the real axis or unit circle
• Techniques such as the Poisson integral formula, the Cauchy integral formula, or the method of images can be applied to solve the transformed problem
• Example: Solving Laplace's equation with Dirichlet boundary conditions in a polygonal domain

Applications of Schwarz-Christoffel Transformation

• The Schwarz-Christoffel transformation is particularly useful for problems involving Laplace's equation, such as electrostatics, fluid flow, and heat conduction, where the geometry of the domain plays a crucial role in determining the solution
• In electrostatics, the transformation can be used to find the electric potential and field in polygonal conductors
• In fluid dynamics, the transformation can be applied to study the flow around polygonal obstacles or through polygonal channels
• In heat conduction, the transformation can be employed to analyze the temperature distribution in polygonal plates or rods
• Example: Determining the electric field around a polygonal conductor with a fixed potential on its boundary