Skip to main content
SpringerLink
Log in

Parabolic Trigonometry

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Different forms of trigonometry have been proposed in the past to account for geometrical and applicative issues. Along with circular trigonometry, its hyperbolic counterpart has played a pivotal role to provide the geometrical framework of special relativity. The parabolic trigonometry is in between the previous two, and we discuss the relevant properties, point out the analogies with the standard forms and to the elementary problem of the projectile parabolic motion.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. It is also easy to infer the derivatives of the inverse TPF functions, namely \(\begin{array}{l} {\frac{d}{dx} {}_{p} c^{-1} (x)=-(1+x^{2} ),} \\ {\frac{d}{dx} {}_{p} s^{-1} (x)=\frac{2-x}{2\sqrt{1-x} } } \end{array}\)

References

  1. Yaglom, I.M.: A Simple Non-euclidean Geometry and Its Physical Basis. Springer, Berlin (1979)

    MATH  Google Scholar 

  2. Davis, H.T.: Introduction to Non-linear Differential and Integral Equations. Dover Publications, New York (1962)

    Google Scholar 

  3. Harkin, A., Harkin, J.B.: Geometry of generalized complex numbers. Math. Mag. 77, 118–129 (2004)

    Article  MathSciNet  Google Scholar 

  4. Sobczyk, G.: New Foundations in Mathematics: The Geometric Concept of Number. Birkhauser, Basel (2013)

    Book  Google Scholar 

  5. Ozdemir, M.: Introduction to hybrid numbers. Adv. Appl. Clifford Algebras (2018). https://doi.org/10.1007/s00006-018-0833-3

    Article  MathSciNet  MATH  Google Scholar 

  6. Dattoli, G., Licciardi, S., Sabia, E.: Generalized trigonometric functions and matrix parameterization. Int. J. Appl. Comput. Math. 3(Suppl 1), 115–128 (2017)

    Article  MathSciNet  Google Scholar 

  7. Ferrari, E.: Bollettino. UMI 18B, 933 (1981)

  8. Edmunds, D.E., Gurka, P., Lang, J.: Properties of generalized trigonometric functions. J. Approx. Theor. 164, 47–56 (2012)

    Article  MathSciNet  Google Scholar 

  9. Gielis, J.: The Geometrical Beauty of Plants. Atlantis Press, Paris (2017)

    Book  Google Scholar 

  10. Dattoli, G., Migliorati, M., Quattromini, M., Ricci, P.: The Parabolic Trigonometric Functions. arXiv:1102.1563v1 [math-ph]

  11. Dattoli, G., Migliorati, M., Ricci, P.E.: The Parabolic Trigonometric Functions and Chebyshev Radicals, ENEA Report RT/2007/21/FIM (2007)

  12. Nickalls, R.W.D.: A new approach to solving the cubic: Cardan’s solution revealed. Math. Gaz. 77, 354–359 (1993)

    Article  Google Scholar 

  13. Witula, R., Slota, D.: Cardano’s formula, square roots, Chebyshev polynomials and radicals. J. Math. Anal. Appl. 363, 639–647 (2010)

    Article  MathSciNet  Google Scholar 

  14. Glasser, M.L.: Hypergeometric functions and the trinomial equation. J. Comput. Appl. Math. 118, 169 (2000)

    Article  MathSciNet  Google Scholar 

  15. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publications, New York (1964)

    MATH  Google Scholar 

  16. Goursat, E.A.: Course in Mathematical Analysis: Functions of a Complex Variable and Differential Equations, vol. 2, pp. 106–120. Dover Publications, New York (1959)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ENEA - Centro Ricerche Frascati, Via Enrico Fermi 45, 00044, Frascati, Rome, Italy

    G. Dattoli, E. Di Palma & S. Licciardi

  2. Department of Biosciences Engineering, University of Antwerp, Groenenborgerlaan 171, 2020, Antwerp, Belgium

    J. Gielis

Authors
  1. G. Dattoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Di Palma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Gielis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Licciardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Di Palma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dattoli, G., Di Palma, E., Gielis, J. et al. Parabolic Trigonometry. Int. J. Appl. Comput. Math 6, 37 (2020). https://doi.org/10.1007/s40819-020-0789-6

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-0789-6

Keywords

Search

Navigation