Geometric model of a physical spline

A physical spline is an elastic bar, the cross-sectional dimensions of which are extremely small compared to the length and radius of curvature of its axis. An example of a physical spline is a thin metal ruler. Such a ruler, when going through given points, immediately acquires a “nature-like” shape, characterized by a minimum energy of internal stresses and a minimum average curvature. Finding the equation of an elastic line is a complex mathematical problem that does not have an elementary solution. The article considers polynomial and parametric methods for geometric modeling of a physical spline going through predetermined points. The elastic line of the physical spline is obtained experimentally. It is shown that the polynomial model significantly differs from the experimentally obtained physical spline, which makes it difficult to use cubic polynomials for modeling an elastic line with big deflections. A parameterized model based on Ferguson curves gives high accuracy of approximation if tangents to the elastic line of the physical spline are specified at the base points. The examples of the physical spline modeling with free and restrained ends are considered. In the case of a free spline, the error of the parametric model was 0.4 %, and in the case of a spline with pinched ends, the error was less than 1.5 %. Calculations were performed using the SMath Studio software tool.