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1.6.A. Estimation of the Bell-Approximation Resolution.

Let's consider the simplest case of two parallel segments placed on distance 2*x_0 from each other. The cross-section of the considered picture is shown on fig.5a. On this picture the position of the approximating segment axial line is determined by parameter x. In this case


Where k - number of points in each segment.

Extrema of such measure are determined by zero values of its derivative:


The denominator of resulting expression is positive at any values of entering variables. The numerator can be transformed to the view


This equation has five solutions, one of which is x_1 = 0, and four others :


Two solutions which are determined by a difference of the expressions placed under sign of exterior square root, have imaginary values at any values of entering variables (as x_0> 0, under the initial agreement - see fig.5a). Others two have real values, when


Simplifying this expression, we shall obtain that at D < sqrt(3)*x_0 investigated conformity function has three extremums: one central and two symmetric side ones. It is easy to show, that:


It is obvious, if D < sqrt(3)*x_0, then S"(x)|_0 > 0, i.e. the central extremum is a local minimum of considered conformity function of and, hence, adjacent side exremums are local maximums of this function (see fig.5b). At D > sqrt(3)*x_0, these maximums merge in one (see fig.5c). Positions of maximums depending on a value of parameter D are shown on fig.5d (see the formula for an evaluation of x_2... x_5).