2008/11/16 by Arnaud Chéritat, Chéritat, Arnaud
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #math.CV
paper · pdf · doi:10.48550/arxiv.0811.2601
32 pages
arxiv created 2008/11/16 · openalex publication_date 2008/11/16 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/03
Consider the straightening phi of a Beltrami form that is constant on a square, with the corresponding ellipses having a vertical major axis, and null outside. A generalized Schwarz-Christoffel formula is used to express the inverse of phi. The formula is found by introducing an affine Riemann surface. This formula is used to draw on a computer the image of the square by phi, and practical aspects are discussed. The resulting shapes are shown for different values of the constant dilatation ratio of the ellipses (=major axis/minor axis). The limit when this ratio tends to infinity is surprising. A model of this limit is proposed, produced by an affine surface uniformization.