Constructing conformal maps of triangulated surfaces (Q408258)

The author describes a means of computing the uniformizing conformal map of a triangulated surface whose triangles are realized as Euclidean triangles in \(\mathbb{R}^3\) onto a fundamental domain in the unit disc \(\mathbb{D}\), plane \(\mathbb{C}\) or sphere \(S^2\); that such a mapping exists and is unique up to Möbius transformations was proved by \textit{A. F. Beardon} and \textit{K. Stephenson} [Indiana Univ. Math. J. 39, No. 4, 1383--1425 (1990; Zbl 0797.30008)].NEWLINENEWLINE His approach uses a circle packing technique (see, for example, [\textit{K. Stephenson}, Introduction to circle packing. The theory of discrete analytic functions. Cambridge: Cambridge University Press (2005; Zbl 1074.52008)]) to create a quasiconformal approximation to the conformal map, and then a discrete form of conformal welding to reduce the distortion and converge in the limit to the required conformal map.NEWLINENEWLINE Mappings of triangulated surfaces arise in areas such as conformal brain flattening (see, for example, [\textit{K. Stephenson}, Notices Am. Math. Soc. 50, No. 11, 1376--1388 (2003; Zbl 1047.52016)]).