Concentric tessellation maps and curvature continuous guided surfaces

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 C2 (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 R2 that joins smoothly with a λ-scaled copy of itself, 0<λ<1. The union of λm-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.