• Corpus ID: 13053882

Convergence of the Zipper algorithm for conformal mapping

@article{Marshall2006ConvergenceOT,
  title={Convergence of the Zipper algorithm for conformal mapping},
  author={Donald E. Marshall and Steffen Rohde},
  journal={arXiv: Complex Variables},
  year={2006}
}
In the early 1980's an elementary algorithm for computing conformal maps was discovered by R. K\"uhnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z_0,...,z_n in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve \gamma with z_0,...,z_n \in \gamma. We prove convergence for Jordan regions in the sense of uniformly close boundaries, and give corresponding uniform estimates… 
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References

SHOWING 1-10 OF 19 REFERENCES
A Multipole Method for Schwarz-Christoffel Mapping of Polygons with Thousands of Sides
TLDR
A method is presented for the computation of Schwarz--Christoffel maps to polygons with tens of thousands of vertices by the use of the fast multipole method and Davis's method for solving the parameter problem.
Univalent functions and Teichm?uller space
I Quasiconformal Mappings.- to Chapter I.- 1. Conformal Invariants.- 1.1 Hyperbolic metric.- 1.2 Module of a quadrilateral.- 1.3 Length-area method.- 1.4 Rengel's inequality.- 1.5 Module of a ring
Boundary Behaviour of Conformal Maps
1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve
Numerische Realisierung konformer Abbildungen durch « Interpolation »
Es wird mit numerischen Beispielen ein neues Verfahren zur naherungsweisen konformen Abbildung des Komplementes eines Schlitzes oder des Auseren einer geschlossenen Jordankurve auf das Ausere des
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