Area-Preserving Geometric Hermite Interpolation

TL;DR

This paper introduces a new class of cubic Bezier curves for planar geometric interpolation that exactly preserve area and achieve higher accuracy than standard methods, with efficient computation and optional iterative refinement.

Contribution

It establishes a family of area-preserving cubic Bezier interpolants that are 5th order accurate, extending geometric Hermite interpolation with exact area preservation.

Findings

Existence of 5th order accurate area-preserving curves

Method is computationally efficient with explicit endpoint parametrization

Optional iterative process improves accuracy while maintaining area preservation

Abstract

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic B\'ezier polynomials. We show there exists a family of such curves which are $5^{th}$ order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.