Abstract
For extended Chebyshev spaces spanned by power functions, the blossoms can be expressed by means of Vandermonde type determinants. When the exponents are nonnegative integers, it is possible to use the classical algorithms for polynomial functions after one or several dimension elevation processes. This provides interesting shape parameters.
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References
P.J. Barry, De Boor–Fix dual functionals and algorithms for Tchebysheffian B-splines curves, Constr. Approx. 12 (1996) 385–408.
D. Bister and H. Prautzsch, A new approach to Tchebysheffian B-splines, in: Curves and Surfaces with Applications to CAGD (Vanderbilt University Press, 1997) pp. 35–42.
S. Karlin and W.J. Studden, Tchebysheff Systems (Wiley, New York, 1966).
R. Kulkarni, P.-J. Laurent and M.-L. Mazure, Non affine blossoms and subdivision for Q-splines, in: Mathematical Methods in Computer Aided Geometric Design II (Academic Press, New York, 1992) pp. 367–380.
P.-J. Laurent, M.-L. Mazure and G. Morin, Shape effects with polynomial Chebyshev splines, in: Curves and Surfaces with Applications in CAGD (Vanderbilt University Press, 1997) pp. 255–262.
M.-L. Mazure, Blossoming of Chebyshev splines, in: Mathematical Methods for Curves and Surfaces (Vanderbilt University Press, 1995) pp. 355–364.
M.-L. Mazure, Chebyshev spaces, RR 952M IMAG, Université Joseph Fourier, Grenoble (January 1996).
M.-L. Mazure, Chebyshev blossoming, RR 953M IMAG, Université Joseph Fourier, Grenoble (January 1996).
M.-L. Mazure, Blossoming: a geometric approach, RR 968M IMAG, Université Joseph Fourier, Grenoble (January 1997), to appear in Constr. Approx.
M.-L. Mazure and P.-J. Laurent, Affine and non affine blossoms, in: Computational Geometry (World Scientific, 1993) pp. 201–230.
M.-L. Mazure and P.-J. Laurent, Marsden identities, blossoming and de Boor–Fix formula, in: Advanced Topics in Multivariate Approximation (World Scientific, 1996) pp. 227–242.
M.-L. Mazure and P.-J. Laurent, Piecewise smooth spaces in duality: application to blossoming, RR 969M 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, RR 978M IMAG, Université Joseph Fourier, Grenoble (September 1997), to appear in Math. Modelling Numer. Anal.
M.-L. Mazure and H. Pottmann, Tchebysheff curves, in: Total Positivity and Its Applications (Kluwer Academic, 1996) pp. 187–218.
H. Pottmann, The geometry of Tchebysheffian splines, Comput. Aided Geom. Design 10 (1993) 181–210.
H. Pottmann and M.G. Wagner, Helix splines as an example of affine Tchebysheffian splines, Adv. Comput. Math. 2 (1994) 123–142.
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).
H.-P. Seidel, New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree, Math. Modelling Numer. Anal. 26 (1992) 149–176.
M.G. Wagner and H. Pottmann, Symmetric Tchebysheffian B-splines schemes, in: Curves and Surfaces in Geometric Design (A.K. Peters, 1994).
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Mazure, ML. Vandermonde type determinants and blossoming. Advances in Computational Mathematics 8, 291–315 (1998). https://doi.org/10.1023/A:1018956616288
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DOI: https://doi.org/10.1023/A:1018956616288