Bézier curves that are close to elastica

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 λ-residual eλ, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B´ezier curve has λ-residual below 0.4, which effectivelyimpliesthatthecurveiswithin1%ofitsarc-lengthtoanelasticcurveinthe L2 norm.

Finally wegive 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