Concentric tessellation maps and curvature continuous guided surfaces

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A multi-sided hole in a surface can be filled by a sequence of nested, smoothly joined surface rings. We show how to generate such a sequence so that (i) the resulting surface is C 2 (also in the limit), (ii) the rings consist of standard splines of moderate degree and (iii) the hole filling closely follows the shape of and replaces a guide surface whose good shape is desirable, but whose representation is undesirable. To preserve the shape, the guided rings sample position and higher-order derivatives of the guide surface at parameters defined and weighted by a concentric tessellating map. A concentric tessellating map maps the domains of n patches to an annulus in R 2 that joins smoothly with a λ-scaled copy of itself, 0 < λ < 1 . The union of λ m -scaled copies parametrizes a neighborhood of the origin and the map thereby relates the domains of the surface rings to that of the guide. The approach applies to and is implemented for a variety of splines and layouts including the three-direction box spline (with Δ-sprocket, e.g. Loop layout, at extraordinary points), tensor-product splines (□-sprocket layout, e.g. Catmull–Clark), and polar layout. For different patch types and layout, the approach results in curvature continuous surfaces of degree less or equal 8, less or equal to ( 6 , 6 ) , and as low as ( 4 , 3 ) if we allow geometric continuity.