Description of the set of admissible piecewise linear routes with n turns in the three dimensional case

Mathematics > Optimization and Control [Submitted on 16 Jun 2026] Title:Description of the set of admissible piecewise linear routes with n turns in the three dimensional case View PDFAbstract: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. Submission history From: Viktor Nefedov V.N. [view email][v1] Tue, 16 Jun 2026 10:36:05 UTC (827 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.