Increasing the smoothness of vector and Hermite subdivision schemes
This work addresses the problem of generating smoother limit curves/surfaces in computer-aided geometric design and related fields, providing a theoretical extension of a known scalar technique to more general settings.
The paper presents a method to increase the smoothness of vector and Hermite subdivision schemes by one order, extending the scalar smoothing factor approach to the vector and Hermite cases. The method constructs schemes of arbitrarily high regularity from convergent schemes.
In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component).