Description of the set of admissible piecewise linear routes with n turns

In this study, we consider polygonal lines connecting two given points $A,B\in\mathbb{R}^2$ and consisting of exactly $n+1$ segments (i.e., having $n$ turn points). The absolute value of the turn angle at each interior point is bounded by a given number $\varphi\in(0,\pi/2)$. Under the condition $n\varphi<\pi$, we describe the set to which all interior vertices of such a polygonal line belong. It is proved that for any point $B^{(1)}$ from this set there exists a polygonal line satisfying the given constraints. Based on these results, an explicit formula is obtained that describes the set of all admissible sequences $\bigl(B^{(1)},\ldots,B^{(n)}\bigr)$ of corner points. The resulting description can serve as a foundation for constructing algorithms for enumerating admissible polygonal lines and for solving optimization problems for an objective function that takes into account both the cost of traversing the segments and the cost of making turns.