Open Access Open Access  Restricted Access Subscription Access


Oleg Stelia, Leonid Potapenko, Ihor Sirenko


This paper presents a new method for constructing a third degree parametric spline curve of C1 continuity. Like the Bèzier curve, the proposed curve is constructed and operated by control points. The peculiarity of the proposed algorithm is the assignment of some unknown values of the spline in the control points abscissas, which are based on the conditions of the first derivative continuity of the curve at these points. The position of the touch points, as well as the control points, can be set interactively. Changing of these points positions leads to a change in the curve shape. This allows the user to flexibly adjust the shape of the curve. Systems of algebraic equations with tridiagonal matrix for calculating the coefficients of a spline curve are constructed. Conditions for the existence and uniqueness of such a curve are presented. Examples of the use of the proposed curve, in particular, for monotone data sets, approximation the ellipse and constructing the letter "S" are given.


Bèzier-type curve; parametric spline curve; third-degree curve; monotonicity.

Full Text:



G. Farin, J. Hoschek, M.-S. Kim (Eds.), Handbook of Computer Aided Geometric Design, Elsevier, 2002.

D. D. Hearn, M. P. Baker, W. Carithers, Computer Graphics with OpenGL, 4th Edition, Pearson Education Limited, 2014.

T. W. Sederberg, Computer Aided Geometric Design. Course Notes, 2012. [Online]. Available at:

P.E. Bezier, “How Renault uses numerical control for car body design and tooling,” Proceedings of the Society of Automotive Engineers Congress SAE, Detroit, paper 680010, 1968. DOI:10.4271/680010.

S.-M. Hashemi-Dehkordi, P. P. Valentini, “Comparison between Bezier and Hermite cubic interpolants in elastic spline formulations,” Acta Mechanica, vol. 225, issue 6, pp. 1809-1821, 2013. DOI: 10.1007/s00707-013-1020-1.

G. Farin, “Geometric Hermite interpolation with circular precision,” Computer Aided Des., vol. 40, issue 4, pp. 476-479, 2008. DOI: 10.1016/j.cad.2008.01.003.

R. T. Farouki, “The Bernstein polynomial basis: A centennial retrospective,” Computer Aided Geometric Design, vol. 29, issue 6, pp. 379-419, 2012. DOI: 10.1016/j.cagd.2012.03.001.

R. Levien, C. H. Sequin, “Interpolating splines: Which is the fairest of them all?”, Computer-Aided Design and Applications, vol. 6, issue 1, pp. 91-102, 2009. DOI: 10.3722/cadaps.2009.91-102.

X. Han, “Piecewise quartic polynomial curves with a local shape parameter,” Journal of Computational and Applied Mathematics, vol. 195, issue 1-2, pp. 34-45, 2006. DOI: 10.1016/

L. Ya, “On the shape parameter and constrained modification of GB-spline curves,” Annales Mathematicae et Informaticae, issue 34, pp. 51-59, 2007.

L. Yan, “Adjustable Bezier curves with simple geometric continuity conditions,” Math. Comput. Appl., vol. 21, issue 4, pp. 44, 2016. DOI: 10.3390/mca21040044.

H. Hang, X. Yao, Q. Li, M. Artiles, “Cubic B-spline curves with shape parameter and their applications,” Mathematical Problems in Engineering, vol. 2017, article ID 3962617, 7 pages, 2017. DOI: 10.1155/2017/3962617.

L. Birk, T. L. McCulloch, “Robust generation of constrained B-spline curves based on automatic differentiation and fairness optimization,” Computer Aided Geometric Design, vol. 59, pp. 49-67, 2018.

X.-A. Han, Y. C. Ma, X. L. Huang, “The cubic trigonometric Bezier curve with two shape parameters,” Applied Mathematics Letters, vol. 22, issue 2, pp. 226-231, 2009. DOI: 10.1016/j.aml.2008.03.015.

M. Dube, R. Sharma, “Quadratic nuat B-spline curves with multiple shape parameters,” International Journal of Machine Intelligence, vol. 3, issue 1, pp. 18-24, 2011. DOI: 10.9735/0975-2927.3.1.18-24.

E. Troll, “Constrained modification of the cubic trigonometric Bezier curve with two shape parameters,” Annales Mathematicae et Informaticae, vol. 43, pp. 145-156, 2014.

L. Yan, “Cubic trigonometric nonuniform spline curves and surfaces,” Mathematical Problems in Engineering, vol. 2016, article ID 7067408, 9 pages, 2016. DOI: 10.1155/2016/7067408.

U. Bashir, M. Abbas, J. M. Ali, “The G2 and G2 rational quadratic trigonometric Bezier curve with two shape parameters with applications,” Appl. Math. Comput., vol. 219, issue 20, pp. 10183-10197, 2013.

V. V. Borisenko, “Construction of optimal Bezier splines,” Fundamental and Applied Mathematics, vol. 21, issue 3, pp. 57-72, 2016. (in Russian)

O. Stelia, L. Potapenko, I. Sirenko, “Application of piecewise-cubic functions for constructing a Bezier type curve of C1 smoothness,” Eastern European Journal of Enterprise Technologies, vol. 2, issue 4-92, pp. 46-52, 2018. DOI: 10.15587/1729-4061.2018.128284.

S. L. Kivva, O. B. Stelya, “A parabolic spline on a nonuniform grid,” Journal of Mathematical Sciences, vol. 109, issue 4, pp. 1715-1725, 2002. DOI: 10.1023/A:1014385915935.

H. A. Akima, “New method of interpolation and smooth curve fitting based on local procedures,” Journal of the Association for Computing Machiners, vol. 17, issue 4, pp. 589-602, 1970.

F. N. Fritsch, R. E. Carlson, “Monotone piecewise cubic interpolation,” SIAM Journal of Numerical Analysis, vol. 17, issue 2, pp. 238-246, 1980.

F. Arándiga, D. F. Yáñez, “Third-order accurate monotone cubic Hermite interpolants,” Applied Mathematics Letters, vol. 4, pp. 73-79, 2019. DOI: 10.1016/j.aml.2019.02.012.

R. L. Burden, J. Douglas Faires, Numerical Analysis, 9th Edition, Boston, MA: Brooks/Cole, Cengage Learning, 2011.

A. A. Samarsky, Theory of Difference Schemes, Moscow, Nauka, 1977, 656 p. (in Russian).

M. Floater, “High-order approximation of conic sections by quadratic splines,” Computer Aided Geometric Design, vol. 12, issue 6, pp. 617-637, 1995.

G. V. Sandrakov, “Homogenization of variational inequalities for problems with regular obstacles,” Doklady Akademii Nauk, vol. 397, issue 2, pp. 170-173, 2004.


  • There are currently no refbacks.
hgs yükleme