The Fagnano Triangle Patrolling Problem

TL;DR

This work connects billiard dynamics to optimal patrolling on triangle edges by showing that the orthic (Fagnano) trajectory minimizes the 1-gap patrolling objective for acute triangles. It introduces a 2-gap variant and proves the existence of infinitely many optimal 2-gap cyclic schedules, all 6-periodic and closely tied to the orthic geometry; it also analyzes a greedy, locally optimal patrolling rule, deriving explicit bounds on its performance relative to the 1-gap optimum. Collectively, the results establish that billiard-type (orthic) trajectories are central to optimal patrolling in triangular domains and offer a concrete, provable bridge between continuous geometry and discrete scheduling. The findings suggest extensions to more complex polygons and multi-agent settings, with potential impact on autonomous surveillance, robotics, and distributed monitoring in geometric domains.

Abstract

We investigate a combinatorial optimization problem that involves patrolling the edges of an acute triangle using a unit-speed agent. The goal is to minimize the maximum (1-gap) idle time of any edge, which is defined as the time gap between consecutive visits to that edge. This problem has roots in a centuries-old optimization problem posed by Fagnano in 1775, who sought to determine the inscribed triangle of an acute triangle with the minimum perimeter. It is well-known that the orthic triangle, giving rise to a periodic and cyclic trajectory obeying the laws of geometric optics, is the optimal solution to Fagnano's problem. Such trajectories are known as Fagnano orbits, or more generally as billiard trajectories. We demonstrate that the orthic triangle is also an optimal solution to the patrolling problem. Our main contributions pertain to new connections between billiard trajectories and optimal patrolling schedules in combinatorial optimization. In particular, as an artifact of our arguments, we introduce a novel 2-gap patrolling problem that seeks to minimize the visitation time of objects every three visits. We prove that there exist infinitely many well-structured billiard-type optimal trajectories for this problem, including the orthic trajectory, which has the special property of minimizing the visitation time gap between any two consecutively visited edges. Complementary to that, we also examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling optimization problem. These trajectories result from a greedy algorithm and can be implemented by a computationally primitive mobile agent.

Figures (7)

• Figure 1: Triangle figure supporting the proof of Fagnano's Theorem.
• Figure 2: The orthic channel (stripe enclosed between the red dotted lines) as it is obtained by 5 triangle reflections.
• Figure 3: A sub-orthic trajectory example.
• Figure 4: Two applications of reflections.
• Figure 5: Six iterations of the greedy patrolling schedule that starts from point $p_0$ of edge $BC$.

Theorems & Definitions (26)

• Theorem 3.1: Fagnano's Theorem
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• proof : of Theorem \ref{['thm: optimal 1-gap cyclic 3-periodic']}
• Lemma 4.1
• proof
• Lemma 5.1
• proof
• Lemma 5.2