Stancu Curves in CAGD


  • Adalbert Csaba Hatvany WMF AG, CAD/CAM Systems, D-73309 Geislingen/Steige, Germany.



curve scheme, de Casteljau algorithm, Bézier curve, Pólya curve, Stancu operator
Abstract views: 191


Starting from the one-parameter dependent linear polynomial Stancu operator, we consider the related polynomial curve scheme with one scalar shape parameter. This scheme, called by us the Stancu curve scheme, generalizes in a suitable manner the classical Bernstein-Bézier scheme and provides more design flexibility by means of the shape parameter.


Download data is not yet available.


Barry, Ph. J., Urn Models, Recursive curve schemes and Computer Aided Geometric Design, Ph.D. thesis, Dept. of Math. Univ. of Utah, Salt Lake City, Utah, 1987.

Barry, Ph. J. and Goldman, R. N., Interpolation and approximation of curves and surfaces using Pólya polynomials, Comput. Vision Graphics Image Process., 53, no. 2, pp. 137-148, 1991, DOI:

Barry, Ph. J., Goldman, R. N. and DeRose, T. D., B-splines, Pólya curves and duality, J. Approx. Theory, 65, pp. 3-21, 1991, DOI:

Eggenberger, F. and Pólya G., Über die Statistik Verketteter Vorgänge, Z. Angew. Math. Mech., 1, pp. 279-289, 1923, DOI:

Eisenberg, S. M. and Wood, B., Approximation of analytic functions by Bernstein type operators, J. Approx. Theory, 6, pp. 242-248, 1972, DOI:

Farin, G., Curves and Surfaces for CAGD. A Practical Guide, 3rd ed., Academic Press, 1993.

Farin, G. and Barry, Ph. J., Link between Bézier and Lagrange curve and surface schemes, Computer-Aided Design, 18, pp. 525-528, 1986, DOI:

Mühlbach, G., Verallgemeinerung der Bernstein- und Lagrange-Polynome. Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu, Rev. Roumaine Math. Pures Appl., 15, pp. 1235-1252, 1970.

Goldman, R. N., Markov chains and Computer-Aided Geometric Design: Part I - Problems and Constraints, ACM Trans. Graph., 3, no. 3, pp. 204-222, 1984, DOI:

Goldman, R. N., Markov chains and Computer-Aided Geometric Design: Part II - Examples and subdivision matrices, ACM Trans. Graph., 4, no. 1, pp. 12-40, 1985, DOI:

Goldman, R. N., Pólya's Urn Model and Computer-Aided Geometric Design, SIAM J. Algebraic Discrete Methods, 6, pp. 1-28, 1985, DOI:

Hatvany, A. Cs., A simplicial approach to Stancu curves and surfaces, in preparation.

Hoschek, J. and Lasser, D., Grundlagen der geometrischen Datenverarbeitung, 2nd ed., Teubner, 1992. DOI:

Mastroiani, G. and Occorsio, M. R., Sulle derivate dei polinomi di Stancu, Rend. Accad. Sci. Fis. Mat. Napoli, 4, no. 45, pp. 273-281, 1978.

Mastroiani, G. and Occorsio, M. R., Una generalizaione dall'operatore di Stancu, Rend. Accad. Sci. Fis. Mat. Napoli, 4, no. 45, pp. 495-511, 1978.

Stancu, D. D., Approximations of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 8, pp. 1173-1194, 1968.

Stancu, D. D., Use of probabilistic methods in the theory of uniform approximation of continuous functions, Rev. Roumaine Math. Pures Appl., 14, pp. 673-691, 1969.

Stancu, D. D., Approximation properties of a class of linear positive operators, Studia Univ. Babeş-Bolyai, Cluj, Ser. Math. Mech., 14, pp. 33-38, 1970.

Walz, G., On Generalized Bernstein polynomials in CAGD, Tech. Report 86, Univ. Mannheim, 1988. DOI:

Walz, G., Spline-Funktionen im Komplexen, B. I. - Wissenschaftsverlag, 1991.

Walz, G., Tigonometric Bézier and Stancu polynomials over intervals and triangles, CAGD, 14, pp. 393-397, 1997, DOI:




How to Cite

Hatvany, A. C. (2002). Stancu Curves in CAGD. Rev. Anal. Numér. Théor. Approx., 31(1), 71–87.