Clothoid fitting and geometric Hermite subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

Joint spectral radius and ternary hermite subdivision

In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.

Stirling numbers and Gregory coefficients for the factorization of Hermite subdivision operators

In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order $d$ allows for $d$ factorizations of the subdivision operator with respect to the Gregory operators: a new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the $d$th factorization provides a ‘convergence from contractivity’ method for showing $C^d$-convergence of the associated Hermite subdivision scheme. Gregory operators are derived by explicitly solving a recursion based on the Taylor operator and iterated vector scheme factorizations. The explicit expression of these operators allows one to compute the $d$th factorization directly from the mask of the Hermite scheme. In particular, it is not necessary to compute intermediate factorizations, which simplifies the procedures used up to now.