Bivariate range functions with superior convergence order

TL;DR

This paper introduces new bivariate range functions with cubic and quartic convergence order, leveraging the Cornelius–Lohner framework, and validates their efficiency through Julia implementations.

Contribution

It presents novel range functions with higher convergence order based on interpolation techniques, improving upon traditional quadratic methods.

Findings

New range functions achieve cubic and quartic convergence order.

Julia implementations demonstrate improved efficiency and efficacy.

Experimental results validate the practical performance of the proposed functions.

Abstract

Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), we exploit the Cornelius--Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy.