###

Multivariate refinable Hermite interpolants

###
Bin Han, Thomas P.-Y. Yu and Bruce Piper

## Abstract

We introduce a general definition of
refinable Hermite interpolants and investigate their general
properties. We study also a notion of symmetry of these refinable
interpolants. Results and ideas from the extensive theory of
general refinement equations are applied to obtain results on
refinable Hermite interpolants. The theory developed here is
constructive and yields an easy-to-use construction method for
multivariate refinable Hermite interpolants. Using this method,
several new refinable Hermite interpolants with respect to
different dilation matrices and symmetry groups are constructed
and analyzed.
Some of the Hermite interpolants constructed here
are related to well-known spline interpolation schemes developed
in the computer-aided geometric design community (e.g. the
Powell-Sabin scheme.)
We make some of these connections precise.
A spline connection allows us to determine critical H\"older
regularity in a trivial way (as opposed to the case of general refinable functions,
whose critical H\"older regularity exponents are often difficult to compute.)
While it is often mentioned in published articles
that ``refinable functions are important
for subdivision surfaces in CAGD applications", it is rather
unclear whether an arbitrary refinable function vector
can be meaningfully applied to build free-form subdivision surfaces.
The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic
developments elsewhere, give %, to the best of our knowledge, the first known
an application of vector
refinability to subdivision surfaces. We briefly discuss
several potential advantages
offered by such Hermite subdivision surfaces.

Back to Preprints and Publications