Abstract
In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing osculating frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly simplifies if the curve is a polynomial one. Further, polynomial PB curves of the degree ≥ 7 and rational PB curves of the degree ≥ 6 that possess rational rotation-minimizing osculating frames are derived, and it is shown that no lower degree curves, constructed from quadratic quaternion polynomials, with such a property exist.
Similar content being viewed by others
References
Beltran, J., Monterde, J.: A characterization of quintic helices. J. Comput. Appl. Math. 206(1), 116–121 (2007)
Choi, H.I., Han, C.Y.: Euler-Rodrigues frames on spatial Pythagorean-hodograph curves. Comput. Aided Geom. Design 19(8), 603–620 (2002)
Farouki, R.T.: Pythagorean-hodograph curves: algebra and geometry inseparable. Geometry and Computing, vol. 1. Springer, Berlin (2008)
Farouki, R.T.: Quaternion and Hopf map characterizations for the existence of rational rotation- minimizing frames on quintic space curves. Adv. Comput. Math. 33(3), 331–348 (2010)
Farouki, R.T., Chang, Y.H., Dospra, P., Sakkalis, T.: Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants. Comput. Aided Geome. Des. 30(7), 653–671 (2013)
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)
Farouki, R.T., Giannelli, C., Sampoli, M.L., Sestini, A.: Rotation-minimizing osculating frames. Comput. Aided Geom. Des. 31(1), 27–42 (2014)
Farouki, R.T., Giannelli, C., Sestini, A.: Helical polynomial curves and double Pythagorean hodographs. I. Quaternion and Hopf map representations. J. Symb. Comput. 44(2), 161–179 (2009)
Farouki, R.T., Giannelli, C., Sestini, A.: Helical polynomial curves and double Pythagorean hodographs. II. Enumeration of low-degree curves. J. Symb. Comput. 44(4), 307–332 (2009)
Farouki, R.T., Han, C.Y.: Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Comput. Aided Geom. Des. 20(7), 435–454 (2003)
Farouki, R.T., Sakkalis, T.: Pythagorean hodographs. IBM J. Res. Develop. 34(5), 736–752 (1990)
Farouki, R.T., Sakkalis, T.: Rational rotation-minimizing frames on polynomial space curves of arbitrary degree. Jour. Symb. Comput. 45, 844–856 (2010)
Farouki, R.T., Sakkalis, T.: Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves. Comput. Aided Geom. Design 28(7), 436–445 (2011)
Farouki, R.T., Sakkalis, T.: A complete classification of quintic space curves with rational rotation-minimizing frames. J. Symbolic Comput 47(2), 214–226 (2012)
Farouki, R.T., Šír, Z.: Rational Pythagorean-hodograph space curves. Comput. Aided Geom. Design 28(2), 75–88 (2011)
Fiorot, J.C., Gensane, T.: Characterizations of the set of rational parametric curves with rational offsets. In: Curves and surfaces in geometric design (Chamonix-Mont-Blanc, 1993), pp. 153–160. A K Peters, Wellesley, MA (1994)
Han, C.Y.: Nonexistence of rational rotation-minimizing frames on cubic curves. Comput. Aided Geom. Design 25(4–5), 298–304 (2008)
Jaklič, G., Kozak, J., Krajnc, M., Vitrih, V., Žagar, E.: An approach to geometric interpolation by Pythagorean-hodograph curves. Adv. Comput. Math. 37, 123–150 (2012)
Kozak, J., Krajnc, M., Vitrih, V.: Dual representation of spatial rational Pythagorean-hodograph curves. Comput. Aided Geom. Des. 31(1), 43–56 (2014)
Krajnc, M., Vitrih, V.: Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves. Math. Comput. Simulation 82(9), 1696–1711 (2012)
Mäurer, C., Jüttler, B.: Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Graph. 3(2), 141–159 (1999)
Monterde, J.: A characterization of helical polynomial curves of any degree. Adv. Comput. Math. 30(1), 61–78 (2009)
Pottmann, H.: Applications of the dual Bézier representation of rational curves and surfaces. In: Curves and surfaces in geometric design (Chamonix-Mont-Blanc, 1993), pp. 377–384. A K Peters, Wellesley, MA (1994)
Pottmann, H.: Rational curves and surfaces with rational offsets. Comput. Aided Geom. Design 12(2), 175–192 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Rida T. Farouki
Rights and permissions
About this article
Cite this article
Kozak, J., Krajnc, M. & Vitrih, V. Parametric curves with Pythagorean binormal. Adv Comput Math 41, 813–832 (2015). https://doi.org/10.1007/s10444-014-9387-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-014-9387-7
Keywords
- Pythagorean-hodograph
- Pythagorean-binormal
- Rational curve
- Dual coordinates
- Rotation-minimizing frame
- Osculating frame