### Application of piecewisecubic functions for constructing a Bezier type curve of C1 smoothness

#### Abstract

We have proposed and implemented a new method for constructing a spline curve of third degree, which possesses the properties of both a cubic spline and the Bezier curve. Similar to building the Bezier curves, control points are assigned, whose position affects the shape of the curve. In the proposed approach, the sections of the straight line that connect control points are tangent to the curve that is constructed. The location of touch points can be different, allowing the construction of different curves for one set of control points. A special feature of the proposed method is assigning, in abscissa of control points, some unknown spline values that are found from conditions for the continuity of the first derivatives of the curve at these points. Finding the coefficients of polynomials that make up the curve comes down to solving a system of linear equations with a three-diagonal matrix. The built curve is a piecewise-cubic function, continuous along with its first derivative throughout the entire interval. Conditions were found for any set of control points in the form of inequalities, which parameters of the curve must meet, at which the curve does exists and it is unique. These conditions follow from the requirement for a diagonal advantage of the matrix of the system for determining coefficients of the curve. A series of computational experiments were performed, which showed that the curve effectively inherits the shape assigned by control points. Similar to the Bezier curves, the proposed curve could be used in computer graphics systems and computer systems for technical design, specifically for the creation of fonts, drawings of parts, elements of transportation vehicles' bodies, etc.

#### Keywords

#### Full Text:

PDF#### References

Bézier, P. E. (1968). How Renault Uses Numerical Control for Car Body Design and Tooling. SAE Technical Paper Series. doi: 10.4271/680010

Forrest, A. R. (1972). Interactive Interpolation and Approximation by Bezier Polynomials. The Computer Journal, 15 (1), 71–79. doi: 10.1093/comjnl/15.1.71

Bernshtein, S. N. (1952). Proof of Weierstrass's theorem based on probability theory. In: Collected works. Vol. 1. Moscow: Publishing House of the USSR Academy of Sciences, 105–106.

Kononiuk, A. E. (2013). Discretely continuous mathematics. Book 6. Surfaces. Part 1. Kyiv: Oswita Ukrainy, 578.

Grigor’ev, M. I., Malozemov, V. N., Sergeev, A. N. (2006). Bernstein polynomials and composite Bézier curves. Computational Mathematics and Mathematical Physics, 46 (11), 1872–1881. doi: 10.1134/s0965542506110042

Boor, C. de (1978). A practical guide to splines. Springer, 348.

Hashemi-Dehkordi, S.-M., Valentini, P. P. (2013). Comparison between Bezier and Hermite cubic interpolants in elastic spline formulations. Acta Mechanica, 225 (6), 1809–1821. doi: 10.1007/s00707-013-1020-1

Levien, R., Séquin, C. H. (2009). Interpolating Splines: Which is the fairest of them all? Computer-Aided Design and Applications, 6 (1), 91–102. doi: 10.3722/cadaps.2009.91-102

Han, X. (2006). Piecewise quartic polynomial curves with a local shape parameter. Journal of Computational and Applied Mathematics, 195 (1-2), 34–45. doi: 10.1016/j.cam.2005.07.016

Ya, L. (2007). On the shape parameter and constrained modification of GB-spline curves. Annales Mathematicae et Informaticae, 34, 51–59.

Hang, H., Yao, X., Li, Q., Artiles, M. (2017). Cubic B-Spline Curves with Shape Parameter and Their Applications. Mathematical Problems in Engineering, 2017, 1–7. doi: 10.1155/2017/3962617

Han, X.-A., Ma, Y., Huang, X. (2009). The cubic trigonometric Bézier curve with two shape parameters. Applied Mathematics Letters, 22 (2), 226–231. doi: 10.1016/j.aml.2008.03.015

Dube, M., Sharma, R. (2011). Quadratic nuat B‐spline curves with multiple shape parameters. International Journal of Machine Intelligence, 3 (1), 18–24. doi: 10.9735/0975-2927.3.1.18-24

Troll, E. (2014). Constrained modification of the cubic trigonometric Bézier curve with two shape parameters. Annales Mathematicae et Informaticae, 43, 145–156.

Yan, L. (2016). Cubic Trigonometric Nonuniform Spline Curves and Surfaces. Mathematical Problems in Engineering, 2016, 1–9. doi: 10.1155/2016/7067408

Hearn, D., Baker, P. (2004). Computer Graphics with OpenGL. Prentice Hall.

Gallier, J. (2015). Curves and surfaces in geometric modeling: Theory and algorithms. Philadelphia, 9.

Rogers, D. F. Adams, J. A. (1990). Mathematical elements for computer grafics. New York: McGraw Hill Publishing Company, 239.

Salomon, D. (2006). Curves And Surfaces for Computer Graphics. Springer, 460. doi: 10.1007/0-387-28452-4

Stelya, O. B. (1997). The existence of a parabolic spline. Appl. Mathem., 1, 62–67.

Kivva, S. L., Stelya, O. B. (2001). About one parabolic spline. Computational technologies, 6 (1), 21–31.

Bakhvalov, N. S. (1975). Numerical methods. Moscow: Nauka, 632.

Samarsky, A. A. (1989). Theory of difference schemes. Moscow: Nauka, 616.

#### GOST Style Citations

Bézier P. E. How Renault Uses Numerical Control for Car Body Design and Tooling // SAE Technical Paper Series. 1968. doi: 10.4271/680010

Forrest A. R. Interactive Interpolation and Approximation by Bezier Polynomials // The Computer Journal. 1972. Vol. 15, Issue 1. P. 71–79. doi: 10.1093/comjnl/15.1.71

Bernshtein S. N. Proof of Weierstrass's theorem based on probability theory // In: Collected works. Vol. 1. Moscow: Publishing House of the USSR Academy of Sciences, 1952. P. 105–106.

Kononiuk A. E. Discretely continuous mathematics. Book 6. Surfaces. Part 1. Kyiv: Oswita Ukrainy, 2013. 578 p.

Grigor’ev M. I., Malozemov V. N., Sergeev A. N. Bernstein polynomials and composite Bézier curves // Computational Mathematics and Mathematical Physics. 2006. Vol. 46, Issue 11. P. 1872–1881. doi: 10.1134/s0965542506110042

Boor C. de A practical guide to splines. Springer, 1978. 348 p.

Hashemi-Dehkordi S.-M., Valentini P. P. Comparison between Bezier and Hermite cubic interpolants in elastic spline formulations // Acta Mechanica. 2013. Vol. 225, Issue 6. P. 1809–1821. doi: 10.1007/s00707-013-1020-1

Levien R., Séquin C. H. Interpolating Splines: Which is the fairest of them all? // Computer-Aided Design and Applications. 2009. Vol. 6, Issue 1. P. 91–102. doi: 10.3722/cadaps.2009.91-102

Han X. Piecewise quartic polynomial curves with a local shape parameter // Journal of Computational and Applied Mathematics. 2006. Vol. 195, Issue 1-2. P. 34–45. doi: 10.1016/j.cam.2005.07.016

Ya L. On the shape parameter and constrained modification of GB-spline curves // Annales Mathematicae et Informaticae. 2007. Vol. 34. P. 51–59.

Cubic B-Spline Curves with Shape Parameter and Their Applications / Hang H., Yao X., Li Q., Artiles M. // Mathematical Problems in Engineering. 2017. Vol. 2017. P. 1–7. doi: 10.1155/2017/3962617

Han X.-A., Ma Y., Huang X. The cubic trigonometric Bézier curve with two shape parameters // Applied Mathematics Letters. 2009. Vol. 22, Issue 2. P. 226–231. doi: 10.1016/j.aml.2008.03.015

Dube M., Sharma R. Quadratic nuat B‐spline curves with multiple shape parameters // International Journal of Machine Intelligence. 2011. Vol. 3, Issue 1. P. 18–24. doi: 10.9735/0975-2927.3.1.18-24

Troll E. Constrained modification of the cubic trigonometric Bézier curve with two shape parameters // Annales Mathematicae et Informaticae. 2014. Vol. 43. P. 145–156.

Yan L. Cubic Trigonometric Nonuniform Spline Curves and Surfaces // Mathematical Problems in Engineering. 2016. Vol. 2016. P. 1–9. doi: 10.1155/2016/7067408

Hearn D., Baker P. Computer Graphics with OpenGL. 3rd ed. Prentice Hall, 2004.

Gallier J. Curves and surfaces in geometric modeling: Theory and algorithms. Philadelphia, 2015. P. 9.

Rogers D. F. Adams J. A. Mathematical elements for computer grafics. 2nd ed. New York: McGraw Hill Publishing Company, 1990. 239 p.

Salomon D. Curves And Surfaces for Computer Graphics. Springer, 2006. 460 p. doi: 10.1007/0-387-28452-4

Stelya O. B. The existence of a parabolic spline // Appl. Mathem. 1997. Issue 1. P. 62–67.

Kivva S. L., Stelya O. B. About one parabolic spline // Computational technologies. 2001. Vol. 6, Issue 1. P. 21–31.

Bakhvalov N. S. Numerical methods. Moscow: Nauka, 1975. 632 p.

Samarsky A. A. Theory of difference schemes. 3rd ed. Moscow: Nauka, 1989. 616 p.

Copyright (c) 2018 Oleg Stelia, Leonid Potapenko, Ihor Sirenko

This work is licensed under a Creative Commons Attribution 4.0 International License.

ISSN (print) 1729-3774, ISSN (on-line) 1729-4061