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Uniform trigonometric polynomial B-spline curves. (English) Zbl 1185.41008
Summary: This paper presents a new kind of uniform spline curve, named trigonometric polynomial B-splines, over space Ω=span{sint,cost,t k-3 ,t k-4 ,,t,1} of which k is an arbitrary integer larger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similar properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
MSC:
41A15Spline approximation
42A10Trigonometric approximation
References:
[1]Farin, G., Curves and Surfaces for Computer Aided Geometric Design, New York: Academic Press, 1988, 1–273.
[2]Piegl, L., Tiller, W., The NURBS Book, 2nd ed., Berlin: Springer, 1997.
[3]Pottmann, H., Wagner, M. G., Helix splines as example of affine Tchebycheffian splines, Advan. in Comput. Math., 1994, 2: 123–142. · Zbl 0832.65008 · doi:10.1007/BF02519039
[4]Mainar, E., Peńa, J. M., Sánchez-Reyes, J., Shape preserving alternatives to the rational Bezier model, Computer Aided Geometric Design, 2001, 18: 37–60. · Zbl 0972.68157 · doi:10.1016/S0167-8396(01)00011-5
[5]Pottmann, H., The geometry of Tchebycheffian spines, Computer Aided Geometric Design, 1993, 10: 181–210. · Zbl 0777.41016 · doi:10.1016/0167-8396(93)90036-3
[6]Zhang, J. W., C-curves: an extension of cubic curves, Computer Aided Geometric Design, 1996, 13: 199–217. · Zbl 0900.68405 · doi:10.1016/0167-8396(95)00022-4
[7]Zhang, J. W., Two different forms of C-B-Splines, Computer Aided Geometric Design, 1997, 14: 31–41. · Zbl 0900.68418 · doi:10.1016/S0167-8396(96)00019-2
[8]Mazure, M. L., Chebyshev-Bernstein bases, Computer Aided Geometric Design, 1999, 16: 649–669. · Zbl 0997.65022 · doi:10.1016/S0167-8396(99)00029-1
[9]Wagner, M. G., Pottmann, H., Symmetric Tchebycheffian B-spline schemes, in Curves and Surfaces in Geometric Design (eds. Laurent, P. J., Le Mehaute, A., Schumaker, L. L.), Natick, MA: AK Peters, 1994, 483–490.
[10]Schumaker, L. L., Spline functions: Basic Theory, New York: Wiley, 1981, 363–499.
[11]Piegl, L., Tiller, W., Curve and surface construction using rational B-splines, Computer Aided Design, 1987, 19: 487–498.
[12]Lane, J. M., Riesenfeld, R. F., A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE Transaction on Pattern Analysis and Machine Intelligence, 1980, PAMI-2(1): 35–46. · Zbl 0436.68063 · doi:10.1109/TPAMI.1980.4766968
[13]Gordan, W. J., Riesenfeld, R. F., B-spline curves and surfaces, Computer Aided Geometric Design, 1974, 95–126.
[14]Morin, G., Warren, J., Weimer, H., A subdivision scheme for surfaces of revolution, Computer Aided Geometric Design, 2001, 18: 483–502. · Zbl 0970.68177 · doi:10.1016/S0167-8396(01)00043-7