Adaptive Computation of Curve Lengths Given by Non-differentiable Functions

Abstract

Measurement of the lengths of curves is quite common in solving various problems. If the function that defines the curve is differentiable, then computing the curve length is a relatively simple mathematical operation. In the absence of initial information about the function, it is necessary to apply approximate methods. Which of these methods should be used for a particular function is usually decided by the user. One of the important factors influencing the choice of the method is the available time resource for the preliminary analysis of the function and for the coordination with the initial data that include both the necessary accuracy of the result and the total numerical costs. The article proposes a method based on an a posteriori approach to the problem, where the analysis of the behavior of the function is carried out in the process of an approximate measurement of the length of the curve in a given area. This method became possible thanks to the introduction of an incremental adaptation mechanism that responds to the deviation of the function curve from the broken line approximating it. As a result, the local analysis accepted as a result of the adaptation made it possible to pass the large steepness segments of the curve in small increments and the flat segments, with large ones. With a particularly sharp change in the function (for example, in sub-domains with singularities), the main adaptation mechanism is able to go beyond the boundaries of the adopted set of constants without serious complications of the algorithm. Thus, there has disappeared the need both for a preliminary analysis of the behavior of the function, not necessarily regular, and the identification of singularities (kinks, extreme points, etc.), their numbers and locations. In order to compute the length of the curve, it is enough to set the function on this area and the required accuracy, limited by the minimum increment, without worrying about using some auxiliary tables and weight factors. The numerical experiment conducted on a test set of functions of varying complexity showed the advantage of the proposed approach over grid methods, especially with equally spaced nodes.

Keywords

• non-differentiable function
• piecewise linear approximation
• adaptive peace-wise selection of nodes
• efficiency index