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

Authors

DOI:

https://doi.org/10.15587/1729-4061.2018.128284

Keywords:

spline curve of third degree, Bezier curve, shape parameters of curve

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.

Author Biographies

Oleg Stelia, Тaras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

PhD, Associate Professor, Senior Researcher

Laboratory of the Computational Methods in the Mechanics of Continuous Media

Leonid Potapenko, Тaras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

PhD

Laboratory of the Computational Methods in the Mechanics of Continuous Media

Ihor Sirenko, Тaras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

Laboratory of the Computational Methods in the Mechanics of Continuous Media

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Published

2018-04-10

How to Cite

Stelia, O., Potapenko, L., & Sirenko, I. (2018). Application of piecewise­cubic functions for constructing a Bezier type curve of C1 smoothness. Eastern-European Journal of Enterprise Technologies, 2(4 (92), 46–52. https://doi.org/10.15587/1729-4061.2018.128284

Issue

Vol. 2 No. 4 (92) (2018): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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

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