Geometric Hermite Interpolation in $\mathbb{R}^n$ by Refinements

TL;DR

This paper introduces a new high-dimensional curve approximation method based on geometric Hermite data, utilizing a novel Hermite average and subdivision scheme that converges to interpolate and approximate sampled curves.

Contribution

It presents a general approach for high-dimensional geometric Hermite interpolation, including a new Hermite average and a convergence proof for the subdivision scheme.

Findings

The subdivision scheme converges to the interpolating curve.

The method effectively approximates high-dimensional curves.

Numerical examples demonstrate the approach's advantages.

Abstract

We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average - a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach.