Description of the set of admissible piecewise linear routes with n turns in the three dimensional case
We consider piecewise linear polygonal chains connecting two given points $A, B \in \mathbb{R}^3$ and consisting of exactly $n+1$ segments (i.e., having $n$ turning points). The absolute value of the turning angle at each interior point is bounded by a given number $\varphi \in (0,\pi)$. Under the condition $n\varphi \leq \pi$, we describe the set to which all interior vertices of such a polygonal chain belong (Theorem 1). It is proved that for any point $B^{(1)}$ from this set, there exists a polygonal chain with the specified parameters (Lemma 1). Based on these results, we obtain an explicit formula describing the set of all admissible sequences $(B^{(1)}, \ldots, B^{(n)})$ of the angular points of the polygonal chain. The obtained description can serve as a basis for constructing algorithms to enumerate admissible polygonal chains and to solve optimization problems for an objective function that accounts for the cost of traversing the segments and the cost of turns.