Approximation of conformal mappings and novel applications to shape recognition of planar domains

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint—a computable approximation to the conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this program where we address the case of simply and doubly connected planar domains. We prove uniform convergence of our approximation scheme to the appropriate conformal mapping. To this end, we affirm a conjecture raised by Ken Stephenson in the 1990s which predicts that the Riemann mapping can be approximated by a sequence of electrical networks. In fact, we first treat a more general case. Consider a planar annulus, i.e., a bounded, 2-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. The resolution of Stephenson’s conjecture then follows by a limiting argument. With more complex topology of the given shape, i.e., when it has higher genus, we will use methods invented by Arabnia (J Parallel Distrib Comput 10:188–192, 1990) and Wani–Arabnia (J Supercomput 25:43–62, 2003). First, to divide the domain into subdomains and thereafter to make the scheme presented in this paper suitable for parallel processing. We will then be able to compare our results for those appearing, for instance, in the work of Arabnia–Oliver (Comput Graph Forum 8:3–11, 1989) that provides algorithms for the translation and scaling of complicated digitalized images.