Parametric curves with Pythagorean binormal

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.

[1]  Hwan Pyo Moon,et al.  Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces , 2002, Adv. Comput. Math..

[2]  Dereck S. Meek,et al.  Hermite interpolation with Tschirnhausen cubic spirals , 1997, Comput. Aided Geom. Des..

[3]  Rida T. Farouki,et al.  Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants , 2013, Comput. Aided Geom. Des..

[4]  Vito Vitrih,et al.  Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curves , 2008, Comput. Aided Geom. Des..

[5]  Rida T. Farouki,et al.  Pythagorean-Hodograph Curves , 2002, Handbook of Computer Aided Geometric Design.

[6]  Vito Vitrih,et al.  On interpolation by Planar cubic G2 pythagorean-hodograph spline curves , 2010, Math. Comput..

[7]  Chang Yong Han.  Geometric Hermite interpolation by monotone helical quintics , 2010, Comput. Aided Geom. Des..

[8]  Rida T. Farouki,et al.  A complete classification of quintic space curves with rational rotation-minimizing frames , 2012, J. Symb. Comput..

[9]  Helmut Pottmann,et al.  Rational curves and surfaces with rational offsets , 1995, Comput. Aided Geom. Des..

[10]  Rida T. Farouki,et al.  The conformal map z -> z2 of the hodograph plane , 1994, Comput. Aided Geom. Des..

[11]  Vito Vitrih,et al.  An approach to geometric interpolation by Pythagorean-hodograph curves , 2011, Advances in Computational Mathematics.

[12]  Rida T. Farouki,et al.  Rational rotation-minimizing frames on polynomial space curves of arbitrary degree , 2010, J. Symb. Comput..

[13]  Zbynek Sír,et al.  Rational Pythagorean-hodograph space curves , 2011, Comput. Aided Geom. Des..

[14]  Rida T. Farouki,et al.  Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable , 2007, Geometry and Computing.

[15]  Rida T. Farouki,et al.  Rotation – minimizing conformal frames , 2013 .

[16]  Rida T. Farouki,et al.  Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves , 2003, Comput. Aided Geom. Des..

[17]  B. Jüttler,et al.  Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics , 1999 .

[18]  Rida T. Farouki,et al.  Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves , 2011, Comput. Aided Geom. Des..

[19]  Helmut Pottmann,et al.  Applications of the dual Be´zier representation of rational curves and surfaces , 1994 .

[20]  T FaroukiRida,et al.  Rational rotation-minimizing frames on polynomial space curves of arbitrary degree , 2010 .

[21]  Juan Monterde,et al.  A characterization of helical polynomial curves of any degree , 2009, Adv. Comput. Math..

[22]  Vito Vitrih,et al.  Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves , 2012, Math. Comput. Simul..

[23]  Rida T. Farouki,et al.  Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations , 2009, J. Symb. Comput..

[24]  J. Fiorot,et al.  Characterizations of the set of rational parametric curves with rational offsets , 1994 .

[25]  T. Sakkalis,et al.  Pythagorean hodographs , 1990 .

[26]  J. Monterde,et al.  A characterization of quintic helices , 2005 .

[27]  Hyeong In Choi,et al.  Euler-Rodrigues frames on spatial Pythagorean-hodograph curves , 2002, Comput. Aided Geom. Des..

[28]  Carla Manni,et al.  Design of rational rotation-minimizing rigid body motions by Hermite interpolation , 2011, Math. Comput..

[29]  Rida T. Farouki,et al.  Rotation-minimizing osculating frames , 2014, Comput. Aided Geom. Des..

[30]  Chang Yong Han.  Nonexistence of rational rotation-minimizing frames on cubic curves , 2008, Comput. Aided Geom. Des..

[31]  Rida T. Farouki,et al.  Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves , 2010, Adv. Comput. Math..

[32]  Vito Vitrih,et al.  Dual representation of spatial rational Pythagorean-hodograph curves , 2014, Comput. Aided Geom. Des..

[33]  Carla Manni,et al.  Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics , 2005, Adv. Comput. Math..

[34]  Rida T. Farouki,et al.  Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves , 2009, J. Symb. Comput..