NATURE-LIKE CURVE MODELING

Abstract

A physical spline is called an elastic rod the cross- section dimensions of which are rather small as compared with the length and radius of its axis curvature. Such a rod when passing through specified points obtains in natural way a nature-like shape characterized with minimum energy of inner stresses and minimum mean curvature. A search for the equation of elastic line is a difficult mathematical problem having no elementary solution. The work purpose: the development of the experimental-rated procedure for modeling a nature-like elastic curve passing through complanar points specified in advance. The investigation methods: methods of piece-cubic interpolation based on the application of polynomial splines and compound curves specified by parametric equations. In the paper there are considered polynomial and parametric methods of the geometric modeling of the physical spline passing through the points specified in advance. The elastic line of the physical spline is obtained experimentally. The investigation results: it is shown that unlike a polynomial model a parametrized model on the basis of Fergusson curve gives high accuracy of approximation if in basic points there are specified tangents to the elastic line of the physical spline with large deflections. Novelty: there is offered a simplified method for the computation of factors of an approximating spline allowing the substitution of the 2n system of nonlinear equations (smoothness conditions) by the successive solution of n systems of two equations. Conclusions: for the modeling of nature-like curves with large deflections there is offered the application of Fergusson cubic spline passing through specified points and touching the specified straight lines in these points. The error of the modeling of the natural elastic line with free ends at n=5 does not exceed 0.4%.