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- W2889062105 abstract "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 a 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. Along the way, we affirm a conjecture raised by Ken Stephenson in the 90's 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, two-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. Our methods involve harmonic mappings and boundary value problems: discrete and analytic in the plane. The scheme we constructed is programmable." @default.
- W2889062105 created "2018-09-07" @default.
- W2889062105 creator A5072571038 @default.
- W2889062105 date "2018-09-01" @default.
- W2889062105 modified "2023-10-18" @default.
- W2889062105 title "Approximation of conformal mappings and novel applications to shape recognition of planar domains" @default.
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- W2889062105 doi "https://doi.org/10.1007/s11227-018-2564-6" @default.
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