Type
Text
Type
Dissertation
Advisor
Bishop, Christopher | Milnor, John | Simons, James | Mullhaupt, Andrew.
Date
2011-12-01
Keywords
conformal mappings, quasiconformal mappings, Schwarz-Christoffel formula | Mathematics
Department
Department of Mathematics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/71026
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
The Riemann Mapping Theorem states that for any proper, simply connected planar domain there exists a conformal mapping from the disk onto the domain. But can this map be explicitly described? For general domains, there is no obvious answer. However, if the domain is the interior of a simple polygon, a convenient formula for the Riemann map was discovered independently by Schwarz and Christoffel. In this dissertation, we present a local quadratically convergent algorithm, the Ahlfors Iteration, based on the theory of quasiconformal maps in the plane, to compute the Schwarz-Christoffel mapping. This algorithm will also apply to a larger collection of simply connected Riemann surfaces. The Ahlfors Iteration improves upon current algorithms that compute the Schwarz-Christoffel map, in that, it is proven to converge, has a simple iterative form, and is easy to implement. | 72 pages
Recommended Citation
Green, Christopher Michael, "The Ahlfors Iteration for Confromal Mapping" (2011). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 233.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/233