### Abstract

A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G^{1} constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.

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

Pages (from-to) | 256-263 |

Number of pages | 8 |

Journal | Computer-Aided Design and Applications |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 4 2018 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computational Mechanics
- Computer Graphics and Computer-Aided Design
- Computational Mathematics

### Cite this

*Computer-Aided Design and Applications*,

*15*(2), 256-263. https://doi.org/10.1080/16864360.2017.1375677

**Fairness metric of plane curves defined with similarity geometry invariants.** / Miura, Kenjiro T.; Suzuki, Sho; Gobithaasan, R. U.; Usuki, Shin; Inoguchi, Jun ichi; Sato, Masayuki; Kajiwara, Kenji; Shimizu, Yasuhiro.

Research output: Contribution to journal › Article

*Computer-Aided Design and Applications*, vol. 15, no. 2, pp. 256-263. https://doi.org/10.1080/16864360.2017.1375677

}

TY - JOUR

T1 - Fairness metric of plane curves defined with similarity geometry invariants

AU - Miura, Kenjiro T.

AU - Suzuki, Sho

AU - Gobithaasan, R. U.

AU - Usuki, Shin

AU - Inoguchi, Jun ichi

AU - Sato, Masayuki

AU - Kajiwara, Kenji

AU - Shimizu, Yasuhiro

PY - 2018/3/4

Y1 - 2018/3/4

N2 - A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.

AB - A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.

UR - http://www.scopus.com/inward/record.url?scp=85032488083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032488083&partnerID=8YFLogxK

U2 - 10.1080/16864360.2017.1375677

DO - 10.1080/16864360.2017.1375677

M3 - Article

AN - SCOPUS:85032488083

VL - 15

SP - 256

EP - 263

JO - Computer-Aided Design and Applications

JF - Computer-Aided Design and Applications

SN - 1686-4360

IS - 2

ER -