Skip to main content

Focal Curves in the Problem of Representing Smooth Geometric Shapes

  • Conference paper
  • First Online:
Advances in Artificial Systems for Medicine and Education IV (AIMEE 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1315))

  • 333 Accesses

Abstract

A focal method for approximating an empirical geometric shape represented by a smooth closed curve is developed. The analytical basis of the method is multifocal lemniscates. The finite number of foci inside the lemniscate and its radius are control parameters of this method. In contrast to the harmonic representation, focal control parameters have of the same nature as the shape itself. In this paper we consider the more general class of multifocal curves from the point of view of using them as a basis for describing geometric shapes. Making the most general requirements for the metric distance function and the invariant description of geometric shapes sets gives a parameterized set of bases in the quasilemniscates class that meet these requirements. Properties of each basis, defined by different functions distances are also reflected in their approximate possibilities. The family multifocal lemniscates is distinguished as the limiting case of parameterized set of basic quasilemniscates. The focal representation of empirical geometric shapes by multifocal lemniscates makes it possible to apply them in different applications.

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

Access this chapter

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

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Hilbert, D.: Gessamelte Abhandlungen, vol. 3, p. 435. Springer, Berlin (1935)

    Google Scholar 

  2. Markushevich, A.I.: The theory of analytical functions, p. 486. T.1 – M., Nauka (1967)

    Google Scholar 

  3. Rakcheeva, T.A.: Multifocus lemniscates: approximation of curves. Comput. Math. Math. Phys. 50(11), 1956–1967 (2010)

    Article  MathSciNet  Google Scholar 

  4. Rakcheeva, T.A.: Focal model in the pattern recognition problem. In: Advances in Intelligent Systems and Computing, vol. 902, pp. 127–138 (2019). https://doi.org/10.1007/978-3-030-12082-5_12

  5. Al-Jubouri, H.A.: Integration colour and texture features for content-based image retrieval. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 12(2), 10–18 (2020). https://doi.org/10.5815/ijmecs.2020.02.02

  6. Mahmoud, S.M., Habeeb, R.S.: Analysis of large set of images using mapreduce framework. Int. J. Mod. Educ. Comput. Sci. (IJMECS), 11(12), 47–52 (2019). https://doi.org/10.5815/ijmecs.2019.12.05

  7. Fahim, A.: A clustering algorithm based on local density of points. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 9(12), 9–16, (2017). https://doi.org/10.5815/ijmecs.2017.12.02

  8. Rakcheeva, T.A.: Quasilemniscates in the task of approximation of the curve forms. Intellect. Syst. 13(1–4), 79–96 (2009)

    Google Scholar 

  9. Rakcheeva, T.A.: Symmetries of polypolar coordination. Vestnik MGOU. Ser. “Phys. Math.” 1, 10–20 (2011)

    Google Scholar 

  10. Rakcheeva, T.A.: Polypolar lemniscate coordinate system. Comput. Res. Model. 1(3), 256–263 (2009)

    Google Scholar 

  11. Gourav, T.S., Singh, H.: Computational approach to image segmentation analysis. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 9(7), 30–37 (2017). https://doi.org/10.5815/ijmecs.2017.07.04

    Article  Google Scholar 

  12. Osaci, M., Cuntan, C.D.: Graphical programming environment for performing physical experiments. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 12(1), 11–17 (2020). https://doi.org/10.5815/ijmecs.2020.01.02

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to T. Rakcheeva .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Rakcheeva, T. (2021). Focal Curves in the Problem of Representing Smooth Geometric Shapes. In: Hu, Z., Petoukhov, S., He, M. (eds) Advances in Artificial Systems for Medicine and Education IV. AIMEE 2020. Advances in Intelligent Systems and Computing, vol 1315. Springer, Cham. https://doi.org/10.1007/978-3-030-67133-4_3

Download citation

Publish with us

Policies and ethics