“Efficient, fair interpolation using Catmull-Clark surfaces” by Halstead, Kass and DeRose

We describe an efficient method for constructing a smooth
surface that interpolates the vertices of a mesh of arbitrary
topological type. Normal vectors can also be interpolated at
an arbitrary subset of the vertices. The method improves
on existing interpolation techniques in that it is fast, robust
and general.
Our approach is to compute a control mesh whose
Catmull-Clark subdivision surface interpolates the given
data and minimizes a smoothness or “fairness” measure of
the surface. Following Celniker and Gossard, the norm we
use is based on a linear combination of thin-plate and membrane energies. Even though Catmull-Clark surfaces do not
possess closed-form parametrizations, we show that the relevant properties of the surfaces can be computed efficiently
and without approximation. In particular, we show that (1)
simple, exact interpolation conditions can be derived, and
(2) the fairness norm and its derivatives can be computed
exactly, without resort to numerical integration.

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