×

Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. (English) Zbl 1069.65551


MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
41A20 Approximation by rational functions
41A63 Multidimensional problems
53A04 Curves in Euclidean and related spaces
65D05 Numerical interpolation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baker, G. A.; Graves-Morris, P., Padé Approximants (1996), University Press: University Press Cambridge
[2] Bangert, C.; Prautzsch, H., Circle and sphere as rational splines, Neural Parallel Sci. Comput., 5, 153-161 (1997) · Zbl 0871.68171
[3] Bishop, R. L., There is more than one way to frame a curve, Amer. Math. Monthly, 82, 246-251 (1975) · Zbl 0298.53001
[4] Brezinski, C.; Van Iseghem, J., Padé approximations, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis III (1994), Elsevier: Elsevier Amsterdam), 47-222 · Zbl 0875.65025
[5] Choi, H. I.; Han, C. Y., Euler-Rodrigues frames on spatial Pythagorean-hodograph curves, Computer Aided Geometric Design, 19, 603-620 (2002) · Zbl 1043.53005
[6] Choi, H. I.; Lee, D. S.; Moon, H. P., Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comp. Math., 17, 5-48 (2002) · Zbl 0998.65024
[7] Chou, J. J., Higher order Bézier circles, Computer-Aided Design, 27, 303-309 (1995) · Zbl 0827.65015
[8] Cuyt, A., Rational Hermite interpolation in one and more variables, (Singh, S. P., Approximation Theory, Spline Functions and Applications (1992), Kluwer Academic: Kluwer Academic Dordrecht), 69-103 · Zbl 0751.41013
[9] Dietz, R.; Hoschek, J.; Jüttler, B., An algebraic approach to curves and surfaces on the sphere and on other quadrics, Computer Aided Geometric Design, 10, 211-229 (1993) · Zbl 0781.65009
[10] Dill, E. H., Kirchhoff theory of rods, Arch. Hist. Exact Sci., 44, 1-23 (1992) · Zbl 0762.01012
[11] Farouki, R. T., The elastic bending energy of Pythagorean hodograph curves, Computer Aided Geometric Design, 13, 227-241 (1996) · Zbl 0875.68861
[12] Farouki, R. T., Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves, Graph. Models, 64, 382-395 (2002) · Zbl 1055.68124
[13] Farouki, R. T.; Sakkalis, T., Pythagorean-hodograph space curves, Adv. Comp. Math., 2, 41-66 (1994) · Zbl 0829.65011
[14] Farouki, R. T.; Shah, S., Real-time CNC interpolators for Pythagorean-hodograph curves, Computer Aided Geometric Design, 13, 583-600 (1996) · Zbl 0875.68875
[15] Farouki, R. T.; al-Kandari, M.; Sakkalis, T., Structural invariance of spatial Pythagorean hodographs, Computer Aided Geometry Design, 19, 395-407 (2002)
[16] Farouki, R. T.; al-Kandari, M.; Sakkalis, T., Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves, Adv. Comp. Math., 17, 369-383 (2002) · Zbl 1001.41003
[17] Farouki, R.T., Han, C.Y., Manni, C., Sestini, A., 2003. Characterization and construction of helical Pythagorean-hodograph quintic space curves. Preprint; Farouki, R.T., Han, C.Y., Manni, C., Sestini, A., 2003. Characterization and construction of helical Pythagorean-hodograph quintic space curves. Preprint · Zbl 1059.65016
[18] Farouki, R. T.; Manjunathaiah, J.; Nicholas, D.; Yuan, G.-F.; Jee, S., Variable feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves, Computer-Aided Design, 30, 631-640 (1998) · Zbl 1049.68723
[19] Fiorot, J.-C.; Jeannin, P.; Cattiaux-Huillard, I., The circle as a smoothly joined BR-curve on [0,1], Computer Aided Geometric Design, 14, 313-323 (1997) · Zbl 0906.68165
[20] Guggenheimer, H., Computing frames along a trajectory, Computer Aided Geometric Design, 6, 77-78 (1989) · Zbl 0664.65017
[21] Jüttler, B., Generating rational frames of space curves via Hermite interpolation with Pythagorean hodograph cubic splines, (Geometric Modeling and Processing ’98 (1998), Bookplus Press), 83-106
[22] Jüttler, B.; Mäurer, C., Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling, Computer-Aided Design, 31, 73-83 (1999) · Zbl 1054.68748
[23] Jüttler, B.; Mäurer, C., Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics, J. Geom. Graphics, 3, 141-159 (1999) · Zbl 0976.53003
[24] Klok, F., Two moving coordinate frames for sweeping along a 3D trajectory, Computer Aided Geometric Design, 3, 217-229 (1986) · Zbl 0631.65145
[25] Kreyszig, E., Differential Geometry (1959), University of Toronto Press: University of Toronto Press Toronto · Zbl 0088.13901
[26] Landau, L. D.; Lifshitz, E. M., Theory of Elasticity (1986), Pergamon Press: Pergamon Press Oxford · Zbl 0178.28704
[27] Lembo, M., On the free shapes of elastic rods, Eur. J. Mech. A Solids, 20, 469-483 (2001) · Zbl 1029.74030
[28] Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (1944), Dover: Dover New York, (reprint) · Zbl 0063.03651
[29] Piegl, L.; Tiller, W., A menagerie of rational B-spline circles, IEEE Comput. Graph. Appl., 9, 5, 48-56 (1989)
[30] Steigmann, D. J.; Faulkner, M. G., Variational theory for spatial rods, J. Elast., 33, 1-26 (1993) · Zbl 0801.73039
[31] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1992), Springer-Verlag: Springer-Verlag New York · Zbl 1004.65001
[32] Struik, D. J., Lectures on Classical Differential Geometry (1988), Dover Publications: Dover Publications New York, (reprint) · Zbl 0041.48603
[33] Tsai, Y.-F.; Farouki, R. T.; Feldman, B., Performance analysis of CNC interpolators for time-dependent feedrates along PH curves, Computer Aided Geometric Design, 18, 245-265 (2001) · Zbl 0971.68172
[34] Uspensky, J. V., Theory of Equations (1948), McGraw-Hill: McGraw-Hill New York · Zbl 0005.11104
[35] Wagner, M. G.; Ravani, B., Curves with rational Frenet-Serret motion, Computer Aided Geometric Design, 15, 79-101 (1997) · Zbl 0894.68150
[36] Wallner, J.; Pottmann, H., Rational blending surfaces between quadrics, Computer Aided Geometric Design, 14, 407-419 (1997) · Zbl 0896.65010
[37] Wang, W.; Joe, B., Robust computation of the rotation minimizing frame for sweep surface modelling, Computer-Aided Design, 29, 379-391 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.