Bézier curves that are close to elastica
Abstract
We study the problem of identifying those cubic Bézier 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 subclass of special Bézier curves as a proxy. We identify an easily computable quantity, which we call the lambdaresidual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a Bézier curve has lambdaresidual below 0.4, which effectively implies that the curve is within 1 percent of its arclength to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input Bézier curve and adjust its length and shape, whilst keeping the endpoints and endtangent angles fixed, until it is close to an elastic curve.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.09192
 Bibcode:
 2017arXiv171009192B
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Graphics
 EPrint:
 13 pages, 15 figures