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Generalization of log-aesthetic curves via similarity geometry


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.

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Both pictures were adapted from [29]

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  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)

    MATH  Google 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)

  3. 3.

    Chou, K.-S., Qu, C.-Z.: Integrable equations arising from motions of plane curves. Phys. D 162, 9–33 (2002)

    MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google 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)

  10. 10.

    Kronrod, A.: Integration with control of accuracy (in Russian). Dokl. Akademii Nauk SSSR 154, 283–286 (1964)

    MathSciNet  Google 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)

    Article  MATH  Google Scholar 

  12. 12.

    Miura, K.T.: A general equation of aesthetic curves and its self-affinity. Comput. Aided Des. Appl. 3, 457–464 (2006)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  Google 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)

  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)

    Article  MATH  Google Scholar 

  18. 18.

    Osada, H., Kotani, S.: Propagation of chaos for the Burgers equation. J. Math. Soc. Jpn. 37(2), 275–294 (1985)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Sato, M., Shimizu, Y.: Generalization of log-aesthetic curves by Hamiltonian formalism. JSIAM Lett. 8, 49–52 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Struik, D.J.: Lectures on Classical Differential Geometry, 2nd edn. Dover, New York (1988)

    MATH  Google 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)

    Book  MATH  Google 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)

    Article  Google 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)

  27. 27.

    Yoshida, N., Saito, T.: Quadratic log-aesthetic curves. Comput. Aided Des. Appl. 14(2), 219–226 (2017)

    Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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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.

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Correspondence to Jun-ichi Inoguchi.

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Inoguchi, Ji., Ziatdinov, R. & Miura, K.T. Generalization of log-aesthetic curves via similarity geometry. Japan J. Indust. Appl. Math. 36, 239–259 (2019).

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

Mathematics Subject Classification

  • 65D17
  • 68U07
  • 53A35
  • 53A04