ConformalMaps is a package for approximating the Riemann map from a simply connected planar domain to a disk. It uses the zipper algorithm as described in Convergence of the zipper algorithm for conformal mapping by Don Marshall and Steffen Rohde.

The domain (approximated by a polygon) is specified as an array which lists the vertices of the domain in counterclockwise order. A conformal map from domain to the unit disk which maps center to the origin is initialized as ConformalMap(domain,center). The keyword argument pointspacing=ϵ inserts equally spaced points along each side of the polygon so that the spacing between consecutive points is everywhere less than ϵ. Smaller values of ϵ give greater accuracy but require longer to compute. The default value is 1% of the diameter of the domain.

using AsyPlots, ConformalMaps
vertices = [1.0  0.0;
            0.0  1.0;
           -1.0  0.0;
            0.0 -1.0]
f = ConformalMap(vertices,0.0)

f supports function call syntax: f(0.1im)

The inverse of f is obtained as inv(f) and is of type InverseConformalMap.

If AsyPlots is installed, then visualize may be used to display the images of a hyperbolic tiling of the disk (if called on an InverseConformalMap) or grid lines (if called on a ConformalMap).

g = inv(f)

Conformal map

visualize returns a ConformalMapVisualization object, whose fields domain and range contain AsyPlots.Plot2D objects. combine returns a single plot with the domain and the range

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