Convergence of a Variant of the Zipper Algorithm for Conformal Mapping

Abstract

In the early 1980s an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z 0 , . . . , zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve γ with z 0 , . . . , zn ∈ γ. We prove convergence for Jordan regions in the sense of uniformly close boundaries and give corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a C 1 curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation. Introduction.Conformal maps have applications to problems in physics, engineering, and mathematics, but how do you find a conformal map, say, of the upper-half plane H to a complicated region? Rather few maps can be given explicitly by hand, so that a computer must be used to find the map approximately. One reasonable way to describe a region numerically is to give a large number of points on the boundary (see Figure 1). One way to say that a computed map defined on H is "close" to a map to the region is to require that the boundary of the image be uniformly close to the polygonal curve through the data points. Indeed, the only information we may have about the boundary of a region is this collection of data points.In the early 1980s an elementary algorithm was discovered independently by Kühnau [K] and the first author. The algorithm is fast and accurate, but convergence was not known. The purpose of this paper is to prove convergence in the sense of uniformly close boundaries and discuss related numerical issues. In many applications both the conformal map and its inverse are required. One important aspect of the algorithm that sets it apart from others is that this algorithm finds both maps simultaneously.The algorithm can be viewed as an approximate solution to a conformal welding problem or as a discretization of the Loewner differential equation. The approximation to the conformal map is obtained as a composition of conformal maps onto slit half planes. Osculation methods also approximate a conformal map by repeated composition of simple maps. See Henrici [H] for a discussion of osculation methods and uniform convergence on compact sets. The algorithms of the present article follow the boundary of a given region much more closely than, for instance, the Koebe algorithm and give uniform convergence on all of H rather than just on compact subsets. Uniform convergence on the closure of the region is particularly important in applications

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

“…(The term "zipper" in the Loewner evolution context has been used before; see the "zipper algorithm" for numerically computing conformal mappings in [MR07] and the references therein.) When t > 0, applying Z CAP t is called "zipping up" the pair of quantum surfaces by t capacity units and applying Z CAP −t is called "zipping down" or "unzipping" by t capacity units.…”

Conformal invariance in two-dimensional turbulence

“…To determine which driving function ξ(t) can generate such a curve, one needs to find the sequence of conformal maps g t (z) that map the half-plane H minus the curve into H itself. We approximate g t (z) by a composition of discrete, conformal slit maps that swallow one segment of the curve at a time (a slight variation of the techniques presented in [26]). This results in a sequence of "times" t i and driving values ξ i that approximate the true driving functions.…”

Computing the Loewner Driving Process of Random Curves in the Half Plane

“…Thus computing the driving function essentially reduces to computing this uniformizing conformal map. We will describe the "zipper algorithm" for doing this [8,13]. Another approach to computing the driving function may be found in [18].…”

“…We use "zipper algorithm" to refer to all the various algorithms we can get from different choices of the curve γ k+1 . Marshall and Rohde [13] use "zipper" to refer only to the choice using tilted slits.…”