Controlled Spline of Third Degree: Approximation Properties and Practical Application

Abstract

The interpolation properties of a Hermitian cubic spline of $$C^{1}$$ smoothness controlled by a broken line, are studied. The angles of inclination of the polygonal line links and linear combinations of values at the vertices of the polygonal line determine the derivatives and spline values in multiple interpolation nodes. Spline nodes are selected to coincide with the abscissas of the vertices of the polygonal line. To calculate the polynomial coefficients of a piecewise polynomial curve, a system of algebraic equations with a three-diagonal matrix is solved. The position of the interpolation nodes plays the role of the shape parameters. When you select spline nodes that match the interpolation nodes, you get a local version of the spline. For the global spline, theoretical estimates of the interpolation error are obtained. The interpolation properties of the considered spline variants are illustrated with various examples. The influence of shape parameters on the behavior of the constructed curve is shown. Some differences between the local and global spline options are shown. An example of using a spline to approximate the electrocardiogram data is given.