Abstract: The construction of space curves with rational rotation- minimizing frames (RRMF curves) by the interpolation of Hermite data, i.e., initial/final points and and frames and , is addressed. Noting that the RRMF quintics form a proper subset of the spatial Pythagorean-hodograph (PH) quintics, characterized by a vector constraint on their quaternion coefficients, and that spatial PH quintic Hermite interpolants possess two free scalar parameters, sufficient degrees of freedom for satisfying the RRMF condition and interpolating the end points and frames can be obtained by relaxing the Hermite data from to . It is shown that, after satisfaction of the RRMF condition, interpolation of the end frames can always be achieved by solving a quadratic equation with a positive discriminant. Three scalar freedoms then remain for interpolation of the end-point displacement , and this can be reduced to computing the real roots of a degree 6 univariate polynomial. The nonlinear dependence of the polynomial coefficients on the prescribed data precludes simple a priori guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely-sampled data from a smooth curve. Modulation of the hodograph by a scalar polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end-point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid-body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.
4.Gerald
Farin, Curves and surfaces for computer-aided geometric
design, 4th ed., Computer Science and Scientific Computing, Academic
Press, Inc., San Diego, CA, 1997. A practical guide; Chapter 1 by P.
Bézier; Chapters 11 and 22 by W. Boehm; With 1 IBM-PC floppy disk
(3.5 inch; HD). MR 1412572
(97e:65022)
11.
R. T. Farouki, C. Giannelli, and A. Sestini (2010), Geometric design using space curves with rational rotation-minimizing frames, in (M. Daehlen et al., eds.), Lecture Notes in Computer Science Vol. 5862, pp. 194-208, Springer.
19.
B. Jüttler and C. Mäurer (1999), Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling, Comput. Aided Design 31, 73-83.
G. Albrecht and R. T. Farouki (1996), Construction of Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comp. Math.5, 417-442. MR 1414289 (97k:65033)
H. I. Choi and C. Y. Han (2002), Euler-Rodrigues frames on spatial Pythagorean-hodograph curves, Comput. Aided Geom. Design 19, 603-620. MR 1937124 (2003i:53002)
H. I. Choi, D. S. Lee, and H. P. Moon (2002), Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comp. Math.17, 5-48. MR 1902534 (2003c:53003)
R. T. Farouki (2010), Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves, Adv. Comp. Math.33, 331-348. MR 2718102
R. T. Farouki, M. al-Kandari, and T. Sakkalis (2002), Structural invariance of spatial Pythagorean hodographs, Comput. Aided Geom. Design 19, 395-407. MR 1917337 (2003g:65019)
R. T. Farouki, M. al-Kandari, and T. Sakkalis (2002), Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves, Adv. Comp. Math.17, 369-383. MR 1916985 (2003e:65011)
R. T. Farouki, C. Giannelli, C. Manni, and A. Sestini (2008), Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures, Comput. Aided Geom. Design 25, 274-297. MR 2408092 (2009c:65020)
R. T. Farouki, C. Giannelli, C. Manni, and A. Sestini (2009), Quintic space curves with rational rotation-minimizing frames, Comput. Aided Geom. Design 26, 580-592. MR 2526013 (2010c:65023)
R. T. Farouki, C. Giannelli, and A. Sestini (2010), Geometric design using space curves with rational rotation-minimizing frames, in (M. Daehlen et al., eds.), Lecture Notes in Computer Science Vol. 5862, pp. 194-208, Springer.
R. T. Farouki and C. Y. Han (2003), Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves, Comput. Aided Geom. Design 20, 435-454. MR 2011551 (2004m:65025)
R. T. Farouki, B. K. Kuspa, C. Manni, and A. Sestini (2001), Efficient solution of the complex quadratic tridiagonal system for PH quintic splines, Numer. Algor. 27, 35-60. MR 1847983 (2002e:65074)
R. T. Farouki and V. T. Rajan (1987), On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4, 191-216. MR 917780 (89a:65028)
R. T. Farouki and T. Sakkalis (2010), Rational rotation-minimizing frames on polynomial space curves of arbitrary degree, J. Symb. Comput. 45, 844-856. MR 2657668
C. Y. Han (2008), Nonexistence of rational rotation-minimizing frames on cubic curves, Comput. Aided Geom. Design 25, 298-304. MR 2408093 (2009a:65035)
B. Jüttler and C. Mäurer (1999), Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling, Comput. Aided Design 31, 73-83.
B. Jüttler and C. Mäurer (1999), Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics, J. Geom. Graphics 3, 141-159. MR 1748024 (2001c:53007)
Rida T. Farouki Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616
Email:
farouki@ucdavis.edu
Carlotta Giannelli Affiliation:
Dipartimento di Sistemi e Informatica, Università degli Studi di Firenze, Viale Morgagni 65, 50134 Firenze, Italy
Email:
giannelli@dsi.unifi.it
Carla Manni Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata,” Via della Ricerca Scientifica, 00133 Roma, Italy
Email:
manni@mat.uniroma2.it
Alessandra Sestini Affiliation:
Dipartimento di Matematica “Ulisse Dini,” Università degli Studi di Firenze, Viale Morgagni 67a, 50134 Firenze, Italy
Email:
alessandra.sestini@unifi.it