Ellipse-preserving Hermite interpolation and subdivision

TL;DR

This paper introduces a new family of Hermite interpolation functions that can exactly reproduce ellipses and exponential polynomials, with proven basis properties and efficient subdivision schemes.

Contribution

It presents a novel family of piecewise-exponential Hermite functions with basis and reproduction properties, along with fast subdivision algorithms for curve interpolation.

Findings

Functions form a Riesz basis

Reproduce exponential polynomials and ellipses

Subdivision schemes are interpolatory and have fourth-order approximation

Abstract

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured.