### Abstract

Original language | English |
---|---|

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

*Computer-Aided Design*,

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

}

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

**Bézier curves that are close to elastica.** / Brander, David; Bærentzen, Jakob Andreas; Fisker, Ann-Sofie; Gravesen, Jens.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Bézier curves that are close to elastica

AU - Brander, David

AU - Bærentzen, Jakob Andreas

AU - Fisker, Ann-Sofie

AU - Gravesen, Jens

PY - 2018

Y1 - 2018

N2 - 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

AB - 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

KW - Cubic Bézier curves

KW - Elastic curves

KW - Splines

KW - Approximation

KW - Computer aided design

KW - Physically-based modeling

U2 - 10.1016/j.cad.2018.05.003

DO - 10.1016/j.cad.2018.05.003

M3 - Journal article

VL - 104

SP - 36

EP - 44

JO - Computer-Aided Design

JF - Computer-Aided Design

SN - 0010-4485

ER -