Advances in Computational Mathematics

, Volume 43, Issue 1, pp 1–24 | Cite as

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

  • Rida T. Farouki
  • Graziano Gentili
  • Carlotta Giannelli
  • Alessandra Sestini
  • Caterina Stoppato


A rotation–minimizing frame (f 1,f 2,f 3) on a space curve r(ξ) defines an orthonormal basis for \(\mathbb {R}^{3}\) in which \(\mathbf {f}_{1}=\mathbf {r}^{\prime }/|\mathbf {r}^{\prime }|\) is the curve tangent, and the normal–plane vectors f 2, f 3 exhibit no instantaneous rotation about f 1. Polynomial curves that admit rational rotation–minimizing frames (or RRMF curves) form a subset of the Pythagorean–hodograph (PH) curves, specified by integrating the form \(\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )\) for some quaternion polynomial \(\mathcal {A}(\xi )\). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial \(\mathcal {A}(\xi )\), a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.


Pythagorean–hodograph curves Rotation–minimizing frames Quaternion polynomials Rotation indicatrix 

Mathematics Subject Classification (2010)

12D05 12Y05 14H45 14H50 53A04 68U05 68U07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barton, M., Jüttler, B., Wang, W.: Construction of rational curves with rational rotation–minimizing frames via Möbius transformations. Mathematical Methods for Curves and Surfaces 2008, Lecture Notes in Computer Science 5862. Springer, Berlin, 15–25 (2010).Google Scholar
  2. Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82, 246–251 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cheng, C.C.-A., Sakkalis, T.: On new types of rational rotation–minimizing space curves. J. Symb. Comput. 74, 400–407 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  4. Choi, H.I., Han, C.Y.: Euler–Rodrigues frames on spatial Pythagorean–hodograph curves. Comput. Aided Geom. Des. 19, 603–620 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  5. Choi, H.I., Lee, D.S., Moon, H.P.: Clifford algebra, spin representation, and rational parameterization of curves and surfaces. Adv. Comp. Math. 17, 5–48 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  6. Damiano, A., Gentili, G., Struppa, D. C.: Computations in the ring of quaternionic polynomials. J. Symb. Comput. 45, 38–45 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  7. Farouki, R.T.: Pythagorean–Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008).CrossRefzbMATHGoogle Scholar
  8. Farouki, R.T.: Quaternion and Hopf map characterizations for the existence of rational rotation–minimizing frames on quintic space curves. Adv. Comp. Math. 33, 331–348 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  9. Farouki, R.T.: Rational rotation–minimizing frames — recent advances and open problems. Appl. Math. Comput. 272, 80–91 (2016).MathSciNetGoogle Scholar
  10. Farouki, R.T., al–Kandari, M., Sakkalis, T.: Structural invariance of spatial Pythagorean hodographs. Comput. Aided Geom. Des. 19, 395–407 (2002).MathSciNetCrossRefGoogle Scholar
  11. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Quintic space curves with rational rotation–minimizing frames. Comput. Aided Geom. Des. 26, 580–592 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  12. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Design of rational rotation–minimizing rigid body motions by Hermite interpolation. Math. Comp. 81, 879–903 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  13. Farouki, R.T., Han, C.Y.: Rational approximation schemes for rotation–minimizing frames on Pythagorean–hodograph curves. Comput. Aided Geom. Des. 20, 435–454 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  14. Farouki, R.T., Han, C.Y., Dospra, P., Sakkalis, T.: Rotation–minimizing Euler–Rodrigues rigid–body motion interpolants. Comput. Aided Geom. Des. 30, 653–671 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  15. Farouki, R.T., Han, C.Y., Manni, C., Sestini, A.: Characterization and construction of helical polynomial space curves. J. Comput. Appl. Math. 162, 365–392 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  16. Farouki, R.T., Sakkalis, T.: Rational rotation–minimizing frames on polynomial space curves of arbitrary degree. J. Symb. Comput. 45, 844–856 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  17. Farouki, R.T., Sakkalis, T.: A complete classification of quintic space curves with rational rotation–minimizing frames. J. Symb. Comput. 47, 214–226 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. Farouki, R.T., Sakkalis, T.: Corrigendum to Rational rotation-minimizing frames on polynomial space curves of arbitrary degree [J. Symbolic Comput. 45 (2010) 844–856]. J. Symb. Comput. 58, 99–102 (2013).CrossRefzbMATHGoogle Scholar
  19. Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Mich. Math. J. 56, 655–667 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  20. Gentili, G., Struppa, D.C.: On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math. 76, 15–25 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  21. Han, C.Y.: Nonexistence of rational rotation–minimizing frames on cubic curves. Comput. Aided Geom. Des. 25, 298–304 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  22. Wang, W., Joe, B.: Robust computation of the rotation–minimizing frame for sweep surface modelling. Comput. Aided Des. 29, 379–391 (1997).CrossRefGoogle Scholar
  23. Wang, W., Jüttler, B., Zheng, D., Liu, Y.: Computation of rotation–minimizing frames. ACM Trans. Graph. 27(1), 1–18 (2008). Article 2.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Rida T. Farouki
    • 1
  • Graziano Gentili
    • 2
  • Carlotta Giannelli
    • 2
  • Alessandra Sestini
    • 2
  • Caterina Stoppato
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaDavisUSA
  2. 2.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly

Personalised recommendations