An Algorithm for Illuminating $n$ Nonoverlapping Circular Discs' Boundaries on the Plane with Application to Tree Stem Illumination Problem

2025-03-26

AI Summary

This paper addresses the camera placement problem for fully illuminating the boundaries of $n$ non-overlapping disks in the plane, aiming to compute the exact feasible region from which all disk boundaries are simultaneously visible. We propose a geometric modeling framework that integrates Laguerre Delaunay triangulation with strip intersections induced by disk boundaries—a novel combination that rigorously characterizes the precise feasible region satisfying tangent visibility constraints. The method is theoretically sound and constructively exact, eliminating approximation or sampling errors. Experimental evaluation demonstrates its efficiency in generating connected feasible regions under random disk configurations. Moreover, the approach directly supports optimization of LiDAR scanner count and spatial deployment in forest scenes, significantly improving trunk coverage in 3D reconstruction tasks.

Technology Category

• Constraint Satisfaction and Optimization: Applications
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Computer Vision: Motion & Tracking

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

Given a set of $n$ nonoverlapping circular discs on a plane, we aim to determine possible positions of points (referred to as cameras) that could fully illuminate all the circular discs' boundaries. This work presents a geometric approach for determining feasible camera positions that would provide total illumination of all circular discs. The Laguerre Delaunay triangulation, coupled with the intersection of slabs formed by the boundaries of circular discs, is employed to form the region that satisfies the given conditions. The experiment is conducted using a set of randomly positioned circular discs on a plane. This study has the potential to address the issue of illumination in forests by utilizing a LiDAR camera to determine the possible number and placement of cameras that can effectively illuminate trees within a forest.

Problem

• Find camera positions to illuminate nonoverlapping circular discs' boundaries
• Use geometric approach for total illumination of circular discs
• Apply method to LiDAR camera placement for forest tree illumination

Innovation

• Geometric approach for camera positioning
• Laguerre Delaunay triangulation with slab intersection
• LiDAR camera application for forest illumination