Abstract
The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials.
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References
P.J. Barry, De Boor-Fix dual functionals and algorithms for Tchebycheffian B-splines curves, Constr. Approx. 12 (1996) 385-408.
D. Bister and H. Prautzsch, A new approach to Tchebycheffian B-splines, in: Curves and Surfaces with Applications to CAGD (Vanderbilt University Press, 1997) pp. 35-42.
N. Dyn and A. Ron, Recurrence relations for Tchebycheffian B-splines, J. Anal. Math. 51 (1988) 118-138.
S. Karlin and W.J. Studden, Tchebycheff Systems (Wiley Interscience, New York, 1966).
T. Lyche, A recurrence relation for Chebyshevian B-splines, Constr. Approx. 1 (1985) 155-173.
M.-L. Mazure, Chebyshev blossoming, RR 953M IMAG, Université Joseph Fourier, Grenoble (January 1996).
M.-L. Mazure, Vandermonde type determinants and blossoming, Adv. Comput. Math. 8 (1998) 291-315.
M.-L. Mazure, Blossoming: a geometrical approach, Constr. Approx. 15 (1999) 33-68.
M.-L. Mazure and P.-J. Laurent, Piecewise smooth spaces in duality: application to blossoming, RR969-M, IMAG, Université Joseph Fourier, Grenoble (January 1997), to appear in J. Approx. Theory.
M.-L. Mazure and P.J. Laurent, Nested sequences of Chebyshev spaces and shape parameters, Math. Modelling Numer. Anal. 32 (1998) 773-788.
M.-L. Mazure and P.J. Laurent, Polynomial Chebyshev splines, RR 998M IMAG, Université Joseph Fourier, Grenoble (January 1998), to appear in Comput. Aided Geom. Design.
M.-L. Mazure and H. Pottmann, Tchebycheff curves, in: Total Positivity and Its Applications (Kluwer Academic, Dordrecht, 1996) pp. 187-218.
H. Pottmann, The geometry of Tchebycheffian splines, Comput. Aided Geom. Design 10 (1993) 181-210.
L. Ramshaw, Blossoming: a connect-the-dots approach to splines, Technical Report, Digital Systems Research Center, Palo Alto, CA (1987).
L. Ramshaw, Blossoms are polar forms, Comput. Aided Geom. Design 6 (1989) 323-358.
L.L. Schumaker, Spline Functions (Wiley Interscience, New York, 1981).
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Mazure, ML. Chebyshev spaces with polynomial blossoms. Advances in Computational Mathematics 10, 219–238 (1999). https://doi.org/10.1023/A:1018995019439
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DOI: https://doi.org/10.1023/A:1018995019439