Advertisement

Generalization of log-aesthetic curves via similarity geometry

  • Jun-ichi InoguchiEmail author
  • Rushan Ziatdinov
  • Kenjiro T. Miura
Original Paper Area 1
  • 21 Downloads

Abstract

The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as \(G^n\) Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation.

Keywords

Log-aesthetic curve Superspiral Similarity geometry Similarity curvature Riccati differential equation 

Mathematics Subject Classification

65D17 68U07 53A35 53A04 

Notes

Acknowledgements

The first author was supported by JSPS KAKENHI, JP15K04834 grant. The authors would like to thank Prof. Rebecca Ramnauth of the Department of Computer Science at Long Island University (USA), who has generously volunteered her valuable time to substantively edit and review this paper. Her care, competence, and conscientiousness are much appreciated. Moreover, the issues, remarks, and very important suggestions of the anonymous reviewers which helped to improve the quality of this paper are appreciated.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Cartan, E.: La Méthode de Repèere Mobile, la Théeorie des Groupes Continus, et les Espaces Généralisés. In: Exposés de Géométrie V, Hermann, Paris (1935)Google Scholar
  3. 3.
    Chou, K.-S., Qu, C.-Z.: Integrable equations arising from motions of plane curves. Phys. D 162, 9–33 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gobithaasan, R.U., Miura, K.T.: Aesthetic spiral for design. Sains Malays. 40, 1301–1305 (2011)Google Scholar
  5. 5.
    Harada, T.: Study of quantitative analysis of the characteristics of a curve. Forma 12(1), 55–63 (1997)Google Scholar
  6. 6.
    Harada, T., Mori, N., Sugiyama, K.: Study of quantitative analysis of the characteristics of a curve (in Japanese). Bull. JSSD 40, 9–16 (1994)Google Scholar
  7. 7.
    Inoguchi, J.: Attractive plane curves in Differential Geometry. In: Mathematical Progress in Expressive Image Synthesis III, pp. 121–135. Springer, Tokyo (2016)Google Scholar
  8. 8.
    Inoguchi, J., Kajiwara, K., Miura, K.T., Sato, M., Schief, W.K., Shimizu, Y.: Log-aesthetic curves as similarity geometric analogue of Eulers elasticae. Comput. Aided Geom. Des. 61, 1–5 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kajiwara, K., Kuroda, T., Matsuura, N.: Isogonal deformation of discrete plane curves and discrete Burgers hierarchy. Pac. J. Math. Ind. 8, Article Number 3, 14 (2016)Google Scholar
  10. 10.
    Kronrod, A.: Integration with control of accuracy (in Russian). Dokl. Akademii Nauk SSSR 154, 283–286 (1964)MathSciNetGoogle Scholar
  11. 11.
    Mineur, Y., Lichah, T., Castelain, J.M., Giaume, H.: A shape controlled fitting method for Beezier curves. Comput. Aided Geom. Des. 15(9), 879–891 (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Miura, K.T.: A general equation of aesthetic curves and its self-affinity. Comput. Aided Des. Appl. 3, 457–464 (2006)CrossRefGoogle Scholar
  13. 13.
    Miura, K.T., Gobithaasan, R.U.: Aesthetic design with log-aesthetic curves and surfaces. Math. Prog. Expressive Image Synth. III, 107–119 (2016)CrossRefzbMATHGoogle Scholar
  14. 14.
    Miura, K.T., Gobithaasan, R.U., Suzuki, S., Usuki, S.: Reformulation of generalized log-aesthetic curves with Bernoulli equations. Comput. Aided Des. Appl. 13(2), 265–269 (2016)CrossRefGoogle Scholar
  15. 15.
    Miura, K.T., Sone, J., Yamashita, A., Kaneko, T.: Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005), Aizu-Wakamutsu, pp. 166–171 (2005)Google Scholar
  16. 16.
    Muftejev, V.G., Ziatdinov, R.: Functionality and aesthetics of curved lines in industrial design: a multi-criteria approach to assessing the quality of forms in CAD systems of the future (in Russian). Vestnik Mashinostroenija 7, 23–27 (2018)Google Scholar
  17. 17.
    Nishinari, K., Takahashi, D.: Analytical properties of ultra-discrete Burgers equation and rule-184 cellular automaton. J. Phys. A Math. Gen. 31, 54395450 (1998)CrossRefzbMATHGoogle Scholar
  18. 18.
    Osada, H., Kotani, S.: Propagation of chaos for the Burgers equation. J. Math. Soc. Jpn. 37(2), 275–294 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pogorelov, A.: Differential Geometry. Nauka, Moscow (1974)Google Scholar
  20. 20.
    Sato, M., Shimizu, Y.: Log-aesthetic curves and Riccati equations from the viewpoint of similarity geometry. JSIAM Lett. 7, 21–24 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sato, M., Shimizu, Y.: Generalization of log-aesthetic curves by Hamiltonian formalism. JSIAM Lett. 8, 49–52 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Struik, D.J.: Lectures on Classical Differential Geometry, 2nd edn. Dover, New York (1988)zbMATHGoogle Scholar
  23. 23.
    Whewell, W.: Of the intrinsic equation of a curve, and its application. Camb. Philos. Trans. 8, 659–671 (1849)Google Scholar
  24. 24.
    Yoshida, M.: Hypergeometric Functions, My Love. Vieweg Verlag, Leipzig (1997)CrossRefzbMATHGoogle Scholar
  25. 25.
    Yoshida, N., Hirakawa, T., Saito, T.: Interactive control of planar class A Bézier curves using logarithmic curvature graphs. Comput. Aided Des. Appl. 5, 121–130 (2008)CrossRefGoogle Scholar
  26. 26.
    Yoshida, N., Saito, T.: On the evolutes of log-aesthetic planar curves (in Japanese). Research Report of College of Industrial Technology. Nihon University, vol. 44, pp. 1–5 (2011)Google Scholar
  27. 27.
    Yoshida, N., Saito, T.: Quadratic log-aesthetic curves. Comput. Aided Des. Appl. 14(2), 219–226 (2017)CrossRefGoogle Scholar
  28. 28.
    Ziatdinov, R.: Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Comput. Aided Geom. Des. 29(7), 510–518 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ziatdinov, R., Yoshida, N., Kim, T.: Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Comput. Aided Geom. Des. 29(2), 129–140 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Jun-ichi Inoguchi
    • 1
    Email author
  • Rushan Ziatdinov
    • 2
  • Kenjiro T. Miura
    • 3
  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan
  2. 2.Department of Industrial and Management EngineeringKeimyung UniversityDaeguRepublic of Korea
  3. 3.Department of Mechanical EngineeringShizuoka UniversityHamamatsuJapan

Personalised recommendations