Schwarz-Christoffel transformation (circular version)
The complex-variables function
maps the closed unit disc in the complex plane conformally onto a polygon with sides, interior angles , and vertices . (The polygon is assumed to be not self-intersecting.) The parameters lie on the unit circle, and depend, generally in a complicated way, on the length of the sides of the polygon.
The fractional powers serve to clamp up an arc of the circle into a pointy angle of measure . Indeed, the proof of the Schwarz-Christoffel formula shows that the function can be decomposed near as
Note that the exponent is — not — because the neigbourhood of a point in the domain space looks like a half-disc. For the same reason, the fractional power used in the formula is to be a single-valued branch continuous on the half-disc. Finally, the extra exponents that appear in the integral representation for come from the power rule for differentiation.
We describe the method used to compute the figure. Points in the domain are first parameterized as , with and ranging over a discrete grid, shown schematically in Figure 4. The integral defining the function is path-independent, and a natural choice for the paths are rays emanating from the origin. When computing the integrals along each ray, we exploit the additivity of the complex path integral:
so that is found by summing a previously-computed value and a new integral to be computed. And the new integral is computed using 32-point Gauss quadrature after reparameterizing the path with .
The computation of the integrand
is straightforward, though we must be careful to respect the branch cuts prescribed above. The function in most computer languages takes a branch cut on the negative axis. To get the single-valued branches we need in this situation, we must instead compute via the expression
where is the direction of the tangent to the circle at the point .
Finally, after having obtained a discrete set of image points traced along each circle , the contours in the figure are obtained by interpolating a curved Bézier spline through the image points.
If a triangle is prescribed with the vertex locations, it is not immediately obvious what the parameters should be to obtain that triangle. In the examples here, we simply avoid this difficulty by arbitrarily choosing the parameters to be equally spaced on the unit circle, and hope that nice figures result.
The parameters are easily determined from the angles of the desired figure; they are, in this example:
The parameters for this star are (rounded to four decimal places):
0.3 Demonstration computer programs
http://svn.gold-saucer.org/repos/PlanetMath/SchwarzChristoffelTransformationCircularVersion/schwarz-christoffel.pyPython source code for producing images of the Schwarz-Christoffel transformation
http://svn.gold-saucer.org/repos/PlanetMath/SchwarzChristoffelTransformationCircularVersion/explanation.pyPython source code for the explanatory diagrams
- 1 Lars V. Ahlfors. Complex Analysis, third edition. McGraw-Hill, 1979.
|Title||Schwarz-Christoffel transformation (circular version)|
|Date of creation||2013-03-22 16:52:57|
|Last modified on||2013-03-22 16:52:57|
|Last modified by||stevecheng (10074)|