B\'ezier curves that are close to elastica

TL;DR

This paper introduces a method to identify cubic Bézier curves that closely approximate elastic curves in the L2 norm, providing geometric criteria and algorithms for shape adjustment while preserving key features.

Contribution

It defines the lambda-residual as a predictor of closeness to elastic curves and develops algorithms to project Bézier curves onto this class with controlled shape modifications.

Findings

Lambda-residual effectively predicts proximity to elastic curves.

Geometric criteria guarantee low lambda-residual for control polygons.

Projection algorithms successfully adjust curves to approximate elastic shapes.

Abstract

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.