Algorithm 952

The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.

[1]  Jirí Kosinka,et al.  A unified Pythagorean hodograph approach to the medial axis transform and offset approximation , 2011, J. Comput. Appl. Math..

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

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

[4]  Carla Manni,et al.  A control polygon scheme for design of planar C2 PH quintic spline curves , 2007, Comput. Aided Geom. Des..

[5]  A. Morgan.  Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems , 1987 .

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

[7]  Helmut Pottmann,et al.  Polynomial and Rational Pythagorean-Hodograph Curves Reconciled , 1994, IMA Conference on the Mathematics of Surfaces.

[8]  Zulfiqar Habib,et al.  G2 Pythagorean hodograph quintic transition between two circles with shape control , 2007, Comput. Aided Geom. Des..

[9]  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..

[10]  Rida T. Farouki,et al.  Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves , 1998, Comput. Aided Des..

[11]  Carla Manni,et al.  Design of C 2 Spatial Pythagorean-Hodograph Quintic Spline Curves by Control Polygons , 2010, Curves and Surfaces.

[12]  Dereck S. Meek,et al.  A generalisation of the Pythagorean hodograph quintic spiral , 2004 .

[13]  Rida T. Farouki,et al.  Construction and shape analysis of PH quintic Hermite interpolants , 2001, Comput. Aided Geom. Des..

[14]  C. Zwikker.  The Advanced Geometry of Plane Curves and Their Applications , 1963 .

[15]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[16]  Carla Manni,et al.  Efficient Solution of the Complex Quadratic Tridiagonal System for C2 PH Quintic Splines , 2001, Numerical Algorithms.

[17]  Carla Manni,et al.  Spatial C^2 PH quintic splines , 2003 .

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

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

[20]  Rida T. Farouki,et al.  Construction ofC2 Pythagorean-hodograph interpolating splines by the homotopy method , 1996, Adv. Comput. Math..

[21]  Rida T. Farouki,et al.  Exact Taylor series coefficients for variable-feedrate CNC curve interpolators , 2001, Comput. Aided Des..

[22]  Carla Manni,et al.  Shape‐preserving interpolation by G1 and G2 PH quintic splines , 2003 .

[23]  C. A. Neff,et al.  Hermite interpolation by Pythagorean hodograph quintics , 1995 .

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

[25]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[26]  Rida T. Farouki,et al.  Structural invariance of spatial Pythagorean hodographs , 2002, Comput. Aided Geom. Des..

[27]  Antonios Tsourdos,et al.  Differential Geometric Path Planning of Multiple UAVs , 2007 .

[28]  Amirmohammad Ghandehariun,et al.  Real-time PH-based interpolation algorithm for high speed CNC machining , 2011 .

[29]  Rida T. Farouki,et al.  G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines , 1999 .

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

[31]  Rida T. Farouki,et al.  Real rational curves are not 'unit speed' , 1991, Comput. Aided Geom. Des..

[32]  Takis Sakkalis,et al.  Pythagorean-hodograph space curves , 1994, Adv. Comput. Math..

[33]  Carla Manni,et al.  Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures , 2008, Comput. Aided Geom. Des..

[34]  Dominiek Reynaerts,et al.  Path planning for mobile and hyper-redundant robots using Pythagorean hodograph curves , 1997, 1997 8th International Conference on Advanced Robotics. Proceedings. ICAR'97.

[35]  Rida T. Farouki,et al.  The elastic bending energy of Pythagorean-hodograph curves , 1996, Comput. Aided Geom. Des..

[36]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[37]  Rida T. Farouki,et al.  On the numerical condition of polynomials in Bernstein form , 1987, Comput. Aided Geom. Des..

[38]  Rida T. Farouki,et al.  Topological Criterion for Selection of Quintic Pythagorean – Hodograph Hermite Interpolants ∗ , 2006 .

[39]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[40]  Rida T. Farouki,et al.  Algorithm 812: BPOLY: An object-oriented library of numerical algorithms for polynomials in Bernstein form , 2001, TOMS.

[41]  Ming-Yang Cheng,et al.  Vision-based Pythagorean hodograph spline command generation and adaptive disturbance compensation for planar contour tracking , 2013 .

[42]  N. Aouf,et al.  3D Cooperative Pythagorean Hodograph path planning and obstacle avoidance for multiple UAVs , 2010, 2010 IEEE 9th International Conference on Cyberntic Intelligent Systems.

[43]  Rida T. Farouki,et al.  Pythagorean-hodograph quintic transition curves of monotone curvature , 1997, Comput. Aided Des..

[44]  D. J. Waltona,et al.  Planar G 2 transition with a fair Pythagorean hodograph quintic curve , 2001 .

[45]  Rida T. Farouki,et al.  Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves , 2002, Adv. Comput. Math..

[46]  Rida T. Farouki,et al.  Real-time CNC interpolators for Pythagorean-hodograph curves , 1996, Comput. Aided Geom. Des..

[47]  Hwan Pyo Moon.  Minkowski Pythagorean hodographs , 1999, Comput. Aided Geom. Des..

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

[49]  Antonios Tsourdos,et al.  Path Planning of UAVs in Urban Region Using Pythagorean Hodograph Curves , 2011 .

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

[51]  Dereck S. Meek,et al.  Planar G 2 transition with a fair Pythagorean hodograph quintic curve , 2002 .

[52]  Gábor Valasek,et al.  Employing Pythagorean Hodograph Curves for Artistic Patterns , 2011, Acta Cybern..

[53]  Kyeong Hah Roh,et al.  Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves , 1999, Comput. Aided Des..

[54]  Mario Fernando Montenegro Campos,et al.  On the Generation of Trajectories for Multiple UAVs in Environments with Obstacles , 2010, J. Intell. Robotic Syst..