## 3 Method
### Method 1: Central Limit Theorem
Note that the longest vectors among the extrema can be located not only on the right tail of the \(S_{n}\) distribution but also on the left tail since the terms \(f^{(i)}(\theta,\varphi)\) can also take negative values and the local minima of the sum \(f(\theta,\varphi)\) can turn out to be negative in value but rather large by the absolute value and, therefore, enter the constructed convex hull based on the known functional property  
\[\max f=-\min(-f).\]  
In the case of a continuous function on a sphere, the extrema of function \(f\) built for object \(O_{1}\) will correspond to the extrema of function \(g\) built for object \(O_{2}\) as rotation of \(O_{1}\). The first vector from the list of the extrema of \(f\) must have the same length as the first vector from \(g\). However, when the function is sampled on a grid, the extrema of continuous functions are not necessarily located precisely at the grid nodes. Therefore, all grid calculations are performed with a certain error, and extrema for \(f\) and \(g\) may differ from each other due to the choice of sampling nodes. This is the reason behind checking several vectors' triplets for the solution and choosing the one with the best correspondence. If a grid on the sphere surfaces is not dense (for example, containing only 100 nodes), the possible error of calculation of the extrema is significant. In such a case, even an exhaustive search on all possible triplets of vectors does not ensure obtaining the correct answer with an acceptable error. For this reason, we use a relatively dense grid. We realize that in the discussion above, our assumption of the independence of \(X_{i}\) is hardly realistic, but at this stage of the presentation, it is only performed to explain the basis of actions, such as why the idea to sum partial reconstruction functions on all the shells has appeared and where the limited number of steps to finalize the analysis could potentially come from. The mentioned results about the trend to normal distribution will not only be valid under the assumption of independence of the random variables \(X_{i}\) (\(X_{i}\) corresponds to the spherical functions \(f_{i}\)), as in [PERSON]'s theorem formulation. In recent decades, [PERSON]'s theorem has also been proven for some classes of dependent random variables, so the trend to normal distribution will also be held for more general cases. Examples of such classes might be the following:  
1. [noitemsep]
2. \(\quad\)Martingale differences [58; 59; 60];
3. \(\quad\)\(m\)-dependent and \(\alpha\)-mixing sequences, such as [54], pp. 387-388, and [61], pp. 773-774, [62; 63; 64];
4. \(\quad\)Mixing conditions [65; 66; 67; 68; 69]. The common feature for all these results is the following. The random variables \"far away\" from each other are almost independent in a certain sense [70]. This happens for the parts of the object under the assumption that these parts weakly depend on each other. There are also classical results by [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] (see [71; 72; 73; 74] (chapter VIII)) for dependent random variables. The formulated conditions of these theorems are either very strict or include conditional distributions, which makes their application difficult. These results are also aligned with the statistical extreme value theory (EVT) [75] (chapter 3) [76]. Both the central limit theorem and the theory of extreme values allow us to estimate the probability of a rare event. This event can be the result of the accumulation of many events or be only one event exceeding some critical threshold [76]. Unfortunately, for these theories, the essential requirement is the identical and independent distribution (i.i.d.) of random variables. The SDF functions corresponding to the spheres cutting the surface are the distance functions. The SDF is continuous by polar angles and the radius. This means that the neighboring spheres at the points with the same angular coordinates cannot differ by an arbitrarily large value.