On Planar Polynomial Geometric Interpolation

TL;DR

This paper revisits planar polynomial geometric interpolation, establishing conditions for existence based on data convexity, and confirms a longstanding conjecture about approximation order for parametric polynomial curves.

Contribution

It provides simple geometric conditions ensuring the existence of interpolants and confirms the H"{o}llig-Koch conjecture for planar parametric polynomial curves.

Findings

Derived sufficient geometric conditions for interpolation existence

Confirmed the H"{o}llig-Koch conjecture in the planar case

Established approximation order for polynomial curves of any degree

Abstract

In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the H\"{o}llig-Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.