Advances in Computational Mathematics

, Volume 45, Issue 1, pp 75–98 | Cite as

Algebraic-Trigonometric Pythagorean-Hodograph space curves

  • Lucia RomaniEmail author
  • Francesca Montagner


We introduce a new class of Pythagorean-Hodograph (PH) space curves - called Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) space curves - that are defined over a six-dimensional space mixing algebraic and trigonometric polynomials. After providing a general definition for this new class of curves, their quaternion representation is introduced and the fundamental properties are discussed. Then, as previously done with their quintic polynomial counterpart, a constructive approach to solve the first-order Hermite interpolation problem in ℝ3 is provided. Comparisons with the polynomial case are illustrated to point out the greater flexibility of ATPH curves with respect to polynomial PH curves.


Space curve Pythagorean-Hodograph Algebraic-Trigonometric Bézier basis Arc length First-order Hermite interpolation 

Mathematics Subject Classification (2010)

65D17 65D05 41A30 41A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carnicer, J.M., Peña, J.M.: Totally positive bases for shape preserving curve design and optimality of B-splines. Comput. Aided Geom. Des. 11, 635–656 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Farouki, R.T., Sakkalis, T.: Pythagorean-hodograph space curves. Adv. Comput. Math. 2, 41–66 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Farouki, R.T., al-Kandari, M., Sakkalis, T.: Hermite interpolation by rotation-invariant spatial pythagorean-hodograph curves. Adv. Comput. Math. 17, 369–383 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput. Aided Geom. Des. 25(4–5), 274–297 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farouki, R.T., Giannelli, C., Sestini, A.: Identification and “reverse engineering” of pythagorean-hodograph curves. Comput. Aided Geom. Des. 34, 21–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    González, C., Albrecht, G., Paluszny, M., Lentini, M.: Design of C 2 algebraic-trigonometric pythagorean-hodograph splines with shape parameters. Comp. Appl. Math. (2016).
  8. 8.
    Kosinka, J., Lávicka, M.: Pythagorean hodograph curves: a survey of recent advances. J. Geom. Graph. 18(1), 23–43 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kozak, J., Krajnc, M., Rogina, M., Vitrih, V.: Pythagorean-hodograph cycloidal curves. J. Numer. Math. 23(4), 345–360 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mainar, E., Peña, J.M.: Corner cutting algorithms associated with optimal shape preserving representations. Comput. Aided Geom. Des. 16, 883–906 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mainar, E., Peña, J.M.: Optimal bases for a class of mixed spaces and their associated spline spaces. Comput. Math. Appl. 59, 1509–1523 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mainar, E., Peña, J.M., Sánchez-Reyes, J.: Shape preserving alternatives to the rational Bézier model. Comput. Aided Geom. Des. 18, 37–60 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Romani, L., Saini, L., Albrecht, G.: Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation. Adv. Comput. Math. 40(5–6), 977–1010 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaMilanoItaly
  2. 2.Mariano Comense (CO)Italy

Personalised recommendations