Application of piecewisecubic functions for constructing a Bezier type curve of C1 smoothness - Stelia, Oleg; Potapenko, Leonid; Sirenko, Ihor (Eastern-European Journal of Enterprise Technologies, 2018)
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.