Abstract
The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional “vertex enclosure constraint.” The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC 1 surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.
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Communicated by Wolfgang Dahmen.
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Peters, J. Smooth interpolation of a mesh of curves. Constr. Approx 7, 221–246 (1991). https://doi.org/10.1007/BF01888155
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DOI: https://doi.org/10.1007/BF01888155