Abstract
Blossoming is a useful technique to study bases and curve representations in computer-aided geometric design. Recently Simeonov et al. (Comput Aided Geom Des 28:549–565, 2011) have used a blossom generalization, namely the h-blossom, to derive new results about the h-Bernstein basis and h-Bézier curves that have previously been studied in approximation theory and computer-aided geometric design. This paper introduces a basis related to the h-Bernstein basis. There is a close relationship between this new basis and the h-Bernstein basis, between the new basis and the h-blossom, and between the new basis and “progressive” curves. This paper explores these relationships and uses them to derive properties both of the new basis itself, and of curves represented in terms of the new basis.
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References
Barry, P., Goldman, R.: Interpolation and approximation of curves and surfaces using generalized Pólya polynomials. Comput. Vis. Graph. Image Process. 53(2), 137–148 (1991)
Barry, P., Goldman, R.: Shape parameter deletion for Pólya curves. Numer. Algorithms 1, 121–138 (1991)
Barry, P., Goldman, R.: Algorithms for progressive curves: extending B-spline and blossoming techniques to the monomial, power, and Newton deal bases. In: Goldman, R., Lyche, T. (eds.) Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces, pp. 11–64. Geometric Design Publications (1993)
Goldman, R.: Pólya’s urn model and computer aided geometric design. SIAM J. Algebr. Discrete Methods 6, 1–28 (1985)
Ramshaw, L.: Blossoming: a connect-the-dots approach to splines, Digital Equipment Corp. Systems Research Center. Technical Report 19 (1987)
Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323–358 (1989)
Simeonov, P., Goldman, R.: Quantum B-splines. BIT Numer. Math. (2012). doi:10.1007/s10543-012-0395-z
Simeonov, P., Zafiris, V., Goldman, R.: h-blossoming: a new approach to algorithms and identities for h-Bernstein bases and h-Bézier curves. Comput. Aided Geom. Des. 28, 549–565 (2011)
Simeonov, P., Zafiris, V., Goldman, R.: q-blossoming: a new approach to algorithms and identities for q-Bernstein bases and q-Bézier curves. J. Approx. Theory 164, 77–104 (2012)
Stancu, D.: Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13, 1173–1194 (1968)
Stefanus, L.Y.: Blossoming off the diagonal. Ph.D. Thesis, University of Waterloo, Waterloo, ON, Canada (1991)
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Barry, P. Properties of an h-Bernstein-like basis and curves. Numer Algor 63, 453–481 (2013). https://doi.org/10.1007/s11075-012-9632-4
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DOI: https://doi.org/10.1007/s11075-012-9632-4