A Curious Case of Curbed Condition

TL;DR

This paper investigates a special family of Bernstein-form polynomials in geometric design, revealing that their evaluation errors are significantly smaller than typical bounds, supported by theoretical proofs and examples.

Contribution

It proves a stronger error bound for a specific polynomial family in Bernstein form, contrasting with general error estimates, and demonstrates this phenomenon through examples.

Findings

Observed errors are smaller than expected bounds for certain polynomials.

Theoretical proof of a stronger error bound for this polynomial family.

Examples illustrate the difference in rounding behavior.

Abstract

In computer aided geometric design a polynomial is usually represented in Bernstein form. The de Casteljau algorithm is the most well-known algorithm for evaluating a polynomial in this form. Evaluation via the de Casteljau algorithm has relative forward error proportional to the condition number of evaluation. However, for a particular family of polynomials, a curious phenomenon occurs: the observed error is much smaller than the expected error bound. We examine this family and prove a much stronger error bound than the one that applies to the general case. Then we provide a few examples to demonstrate the difference in rounding.