C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Abstract The problem of constructing a rational adapted frame ( f 1 ( ξ ) , f 2 ( ξ ) , f 3 ( ξ ) ) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r ( ξ ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C 1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C 2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.

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

[2]  Rida T. Farouki,et al.  Rational frames of minimal twist along space curves under specified boundary conditions , 2018, Adv. Comput. Math..

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

[4]  Bert Jüttler,et al.  Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling , 1999, Comput. Aided Des..

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

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

[7]  Bert Jüttler,et al.  Computation of rotation minimizing frames , 2008, TOGS.

[8]  Rida Farouki.  Rational rotation-minimizing frames on polynomial space curves: recent advances and open problems , 2016 .

[9]  Rida T. Farouki,et al.  A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames , 2017, Adv. Comput. Math..

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

[11]  Rida T. Farouki,et al.  The Bernstein polynomial basis: A centennial retrospective , 2012, Comput. Aided Geom. Des..

[12]  Karla Pockaj.  Hermite G1 rational spline motion of degree six , 2013, Numerical Algorithms.

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

[14]  Vito Vitrih,et al.  Hermite interpolation by rational GK motions of low degree , 2013, J. Comput. Appl. Math..

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

[16]  Bert Jüttler.  Generating rational frames of space curves via hermite interpolation with Pythagorean hodograph cubic splines , 1998 .

[17]  R. Bishop.  There is More than One Way to Frame a Curve , 1975 .

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

[19]  Barry Joe,et al.  Robust computation of the rotation minimizing frame for sweep surface modeling , 1997, Comput. Aided Des..

[20]  Beer S Bhadauria,et al.  Finite Difference Formulae for Unequal Sub- Intervals Using Lagrange's Interpolation Formula , 2009 .

[21]  Bert Jüttler.  Rotation Minimizing Spherical Motions , 1998 .

[22]  Chang Yong Han,et al.  Cubic helical splines with Frenet-frame continuity , 2011, Comput. Aided Geom. Des..

[23]  Rida T. Farouki,et al.  Rational rotation-minimizing frames - Recent advances and open problems , 2016, Appl. Math. Comput..

[24]  Bert Jüttler,et al.  Geometric Interpolation by Quartic Rational Spline Motions , 2010 .

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

[26]  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..

[27]  Bert Jüttler,et al.  Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces , 2005, IMA Conference on the Mathematics of Surfaces.