Skip to main content
SpringerLink
Log in
Book cover

Mathematical Progress in Expressive Image Synthesis III pp 121–135Cite as

Attractive Plane Curves in Differential Geometry

  • Conference paper
  • First Online:

Part of the book series: Mathematics for Industry ((MFI,volume 24))

Abstract

The purpose of chapter is to discuss plane curves from differential geometric point of view and applications of plane curves to computer aided designs. Plane curves are determined uniquely by curvatures up to Euclidean motions. Thus geometry of plane curves are formulated by the Euclidean motion group \(\mathrm {SE}(2)\). From industrial point of view, other transformation groups are more appropriate for characterizing certain classes of plane curves. For instance, under equiaffine transformation group, conics are characterized as plane curves with constant equiaffine curvatures. Plane curves with monotonous curvature function have been paid much attention in industrial shape design and computer aided geometric design. In this chapter we study plane curves with monotonous curvature function, especially log-aesthetic curves, in terms of similarity transformation group.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. J.M. Ali, R.M. Tookey, J.V. Ball, A.A. Ball, The generalised Cornu spiral and its application to span generation. J. Comput. Appl. Math. 102(1), 37–47 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Anjyo, H. Ochiai, Mathematical Basics of Motion and Deformation in Computer Graphics (Morgan & Claypool, San Rafael, 2014)

    Google Scholar 

  3. I. Baran, J. Lehtinen, J. Popović, Sketching clothoid splines using shortest paths. Comp. Graph. Forum 29, 655–664 (2010)

    Article  Google Scholar 

  4. F. Bertails-Descoubes, Super-clothoids. Comput. Graph. Forum 31(2pt2), 509–518 (2012) (Proceedings of the Eurographics’ 12)

    Google Scholar 

  5. F. Bertails-Descoubes, Geometry and mechanics of fibers: some numerical models. in Mathematical Progress in Expressive Image Synthesis III, ed. by Y. Dobashi, H. Ochiai (Springer, Singapore, 2016). doi:10.1007/978-981-10-1076-7_1

    Google Scholar 

  6. E. Cartan, La Méthode de Repèere Mobile, la Thèeorie des Groupes Continus, et les Espaces Géenéralisés, In: Exposés de Géométrie V, Hermann, Paris (1935)

    Google Scholar 

  7. F. Dillen, The classification of hypersurfaces of a Euclidean space with parallel higher order fundamental form. Math. Z. 203, 635–643 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. R.U. Gobithaasan, K.T. Miura, Aesthetic spiral for design. Sains Malaysiana 40, 1301–1305 (2011)

    Google Scholar 

  9. A. Gray, E. Abbena, S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd edn. (CRC Press, Boca Raton, 2006)

    MATH  Google Scholar 

  10. T. Harada, Study of quantitative analysis of the characteristics of a curve. Forma 12(1), 55–63 (1997)

    Google Scholar 

  11. T. Harada, N. Mori, K. Sugiyama, Study of quantitative analysis of curve’s character (in Japanese). Bull. JSSD 40, 9–16 (1994)

    Google Scholar 

  12. J. Inoguchi, Introduction to Klein geometries (Kikagaku Iro Iro, in Japanese) (Nippon Hyoronsha, Tokyo, 2007)

    Google Scholar 

  13. J. Inoguchi, Plane Curves and Solitons (Kyokusen to Soliton, in Japanese) (Asakura Publishing, Tokyo, 2010)

    Google Scholar 

  14. J. Inoguchi, Can We Solve Riccati equations ? (Riccati no Himitsu, in Japanese) (Nippon Hyoronsha, Tokyo, 2010)

    Google Scholar 

  15. K. Kajiwara, T. Kuroda, N. Matsuura, Isogonal deformation of discrete plane curves and discrete Burgers hierarchy. Pac. J. Math. Ind. 8, 3 (2016)

    Google Scholar 

  16. B. Kimia, I. Frankel, A. Popescu, Euler spiral for shape completion. Int. J. Comput. Vision 54, 157–180 (2003)

    Article  MATH  Google Scholar 

  17. J. McCrae, K. Singh, Sketching piecewise clothoid curves. Comput. Graph. 33, 452–461 (2009)

    Article  Google Scholar 

  18. D.S. Meek, D.J. Walton, The use of Cornu spirals in drawing planar curves of controlled curvature. J. Comput. Appl. Math. 25, 69–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  20. K.T. Miura, R.U. Gobithaasan, Aesthetic design with log-aesthetic curves and surfaces. in Mathematical Progress in Expressive Image Synthesis III, ed. by Y. Dobashi, H. Ochiai (Springer, Singapore, 2016). doi:10.1007/978-981-10-1076-7_12

    Google Scholar 

  21. K.T. Miura, J. Sone, A. Yamashita, T. Kaneko, Derivation of a general formula of aesthetic curves, in Proceedings of the 8th International Conference on Humans and Computers (HC2005) (Aizu-Wakamutsu, Japan, 2005), pp. 166–171

    Google Scholar 

  22. K.T. Miura, R.U. Gobithaasan, S. Suzuki, S. Usuki, Reformulation of generalized logaesthetic curves with Bernoulli equations. Comput.-Aided Des. Appl. 13(2), 265–269 (2016)

    Google Scholar 

  23. K. Nomizu, T. Sasaki, Affine Differential Geometry. Geometry of Affine Immersions (Cambridge University Press, Cambridge, 1994)

    MATH  Google Scholar 

  24. H. Ochiai, K. Anjyo, Mathematical formulation of motion and deformation and its applications, Mathematical Progress in Expressive Image Synthesis I, vol. 4, Mathematics for Industry (Springer, Tokyo, 2014), pp. 123–129

    Chapter  Google Scholar 

  25. M. Sato, Y. Shimizu, Log-aesthetic curves and Riccati equations from the viewpoint of similarity geometry. JSIAM Lett. 7, 21–24 (2015)

    Article  MathSciNet  Google Scholar 

  26. A.A. Savelov, in Plane Curves: Systematics, Properties, Applications (in Russian), ed. by A.P. Norden (Gosudarstv. Izdat. Fiz.-Mat. Lit, Moscow, 1960)

    Google Scholar 

  27. N. Yoshida, T. Saito, Interactive aesthetic curve segments. Vis. Comput. 22, 9–11 (2006) (Proceedings of the Pacific Graphics)

    Google Scholar 

  28. N. Yoshida, T. Saito, On the evolutes of log-aesthetic planar curves (in Japanese). Research Report A of College of Industrial Technology. Nihon University 44, 1–5 (2011)

    Google Scholar 

  29. R. Ziatdinov, Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Comput. Aided Geom. Des. 29, 510–518 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Ziatdinov, R.I. Nabiyev, K.T. Miura, MC-curves and aesthetic measurements for pseudospiral curve segments. Math. Des. Tech. Aesthet. 1(1), 6–17 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

The author would like to thank the organizers for inviting him to the workshop MEIS2015. The author would also like to thank Professor Kenjiro T. Miura, Mr. Masayuki Sato, Mr. Yasuhiro Shimizu and Professor Rushan Ziatdinov for their useful comments.

This work is partially supported by KAKENHI 15K04834.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

    Jun-ichi Inoguchi

Authors
  1. Jun-ichi Inoguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun-ichi Inoguchi .

Editor information

Editors and Affiliations

  1. Grad. School of Info. Sci. and Tech, Hokkaido University, Sapporo, Hokkaido, Japan

    Yoshinori Dobashi

  2. Kyushu University, Fukuoka, Japan

    Hiroyuki Ochiai

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this paper

Cite this paper

Inoguchi, Ji. (2016). Attractive Plane Curves in Differential Geometry. In: Dobashi, Y., Ochiai, H. (eds) Mathematical Progress in Expressive Image Synthesis III. Mathematics for Industry, vol 24. Springer, Singapore. https://doi.org/10.1007/978-981-10-1076-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-1076-7_13

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-1075-0

  • Online ISBN: 978-981-10-1076-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation