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Boundary Element Formulation of Harmonic Coordinates

We explain how Boundary Element Methods (BEM) can be used to speed up the computation and reduce the storage associated with Harmonic Coordinates. In addition, BEM formulation allows extending the harmonic coordinates to the exterior and makes possible to compare the transfinite harmonic coordinates with transfinite Shepard interpolation and Mean Value Coordinates. This comparison reveals that there are unifying formulas, yet harmonic coordinates belong to a fundamentally different end of the spectrum. This observation allows us to generalize harmonic coordinates by introducing a novel class of interpolates which we call weakly singular transfinite interpolates. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Boundary representations; Curve, surface, solid and object representations; Geometric Algorithms, languages, and systems; G.1 [Numerical Analysis]: Interpolation; Partial Differential Equations; Integral Equations