Bézier curves that are close to elastica
For designers needing elastic curves in digital environments, this work offers a practical proxy using Bézier curves, though the results are incremental.
The authors identify cubic Bézier curves that are close to planar elastic curves in the L2 norm, using a computable lambda-residual to predict closeness. They provide geometric criteria ensuring the residual is below 0.4, corresponding to less than 1% arc-length difference, and propose projection algorithms to adjust Bézier curves to be near elastic curves.
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 sub-class of special Bézier 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 Bézier 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 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.