Finite approximation methods for two-dimensional sets and their application to geometric optimization problems

This study investigates the problem of approximating closed bounded sets in two-dimensional real space by finite subsets with a given accuracy in the Hausdorff metric. The main focus is on developing an effective approximation method for the class of sets defined by stepwise systems of inequalities.
The proposed method is based on constructing special grid structures that allow controlling the approximation accuracy through a parameter $\tau>0$. Corresponding theoretical statements about the properties of such approximations are proved.
The problem of finding an optimal piecewise-linear path between two points with a single turn under angle constraints is examined in detail. The developed methods are applicable for solving various geometric optimization problems.