Bezier curves that are close to elastica

作者：Brander David^*; Baerentzen Jakob Andreas; Fisker Ann Sofie; Gravesen Jens

来源：Computer-Aided Design, 2018, 104: 36-44.

DOI：10.1016/j.cad.2018.05.003

• Cubic Bezier curves
• Elastic curves
• Splines
• Approximation
• Computer aided design
• Physically-based modeling

We study the problem of identifying those cubic Bezier 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 Bezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual e(lambda), that accurately predicts a small L-2 distance. We then identify geometric criteria on the control polygon that guarantee that a Bezier curve has A-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 Bezier 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.

• 出版日期2018-11