Bézier curves that are close to elastica. (November 2018) - Brander, David; Bærentzen, Jakob Andreas; Fisker, Ann-Sofie; Gravesen, Jens
Abstract: We study the problem of identifying those cubic Bézier curves that are close in the L 2 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 Bézier curves as a proxy. We identify an easily computable quantity, which we call the λ -residual e λ, that accurately predicts a small L 2 distance. We then identify geometric criteria on the control polygon that guarantee that a Bézier curve has λ -residual below 0.4, which effectively implies that the curve is within 1% of its arc-length to an elastic curve in the L 2 norm. Finally we give two projection algorithms that take an input Bézier 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. Highlights: A practical solution to the problem of designing for fabrication methods that create elastic curves. Two easily computable characterizations of elastica-like Bézier curves are found. Fast, reliable algorithms for projecting cubic Bézier curves to elastica-like curves.