### Abstract

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

Original language | English |
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Journal | Computer-Aided Design |

Volume | 104 |

Pages (from-to) | 36-44 |

Number of pages | 9 |

ISSN | 0010-4485 |

DOIs | |

Publication status | Published - 2018 |

### Keywords

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

## Cite this

Brander, D., Bærentzen, J. A., Fisker, A-S., & Gravesen, J. (2018). Bézier curves that are close to elastica.

*Computer-Aided Design*,*104*, 36-44. https://doi.org/10.1016/j.cad.2018.05.003