Problem of Finding an Optimal Piecewise Linear Path Connecting Two Given Points with the Possibility of Making n Turns

TL;DR

This paper addresses the problem of finding an optimal piecewise linear path with a limited number of turns between two points, characterizing feasible regions, and developing algorithms for approximate solutions considering traversal and turn costs.

Contribution

It provides a geometric characterization of feasible interior vertices, explicit formulas for admissible turn sequences, and algorithms for approximate optimization solutions.

Findings

Characterized the region containing all interior vertices of feasible paths.

Derived explicit expressions for admissible turn sequences.

Developed algorithms for approximate path optimization considering costs.

Abstract

We consider the problem of finding an optimal piecewise linear path (polygonal line) connecting two given points with the possibility of making n turns at some points (the absolute value of each turn angle does not exceed a prescribed bound). Under some condition, we characterize the region to which all interior vertices of such a path must belong (Theorem 1). It is shown that for any point from this region, there exists a polygonal line satisfying the given constraints (Lemma 1). Based on these findings, an explicit expression is derived (Theorem 2) that describes the collection of all admissible sequences of corner points. This expression is then used to construct a finite family of sequences that approximates the aforementioned collection. The resulting finite approximating family serves as the basis for developing algorithms that provide approximate solutions to an optimization problem, where the objective function accounts for both the cost of traversing the segments and the cost associated with the turns.