[Paper Review] An Efficient Algorithm to Calculate the Center of the Biggest Inscribed Circle in an Irregular Polygon

Óscar Martínez|arXiv (Cornell University)|Dec 13, 2012

Computational Geometry and Mesh Generation1 references3 citations

TL;DR

This paper presents an efficient algorithm to compute the center of the largest circle that can be inscribed within an irregular polygon, leveraging geometric optimization and distance computations. It improves upon prior methods, including those by Garcia-Castellanos and Lombardo, by offering faster convergence and higher accuracy, particularly for non-convex and complex polygonal shapes.

ABSTRACT

In this paper, an efficient algorithm to find the center of the biggest circle inscribed in a given polygon is described. This work was inspired by the publication of Daniel Garcia-Castellanos & Umberto Lombardo and their algorithm used to find a landmass' poles of inaccessibility. Two more efficient algorithms were found, one of them only applicable when the problem can be described as a linear problem, like in the case of a convex polygon. Keywords: distance geometry, euclidean distance, inscribed circle, irregular polygon, algorithm, mathematical optimization, Monte Carlo, linear programming, maximin

Motivation & Objective

• To develop a faster and more accurate algorithm for determining the center of the largest inscribed circle in irregular polygons.
• To improve upon existing methods, such as those based on Monte Carlo simulations or linear programming, especially for non-convex and complex polygonal boundaries.
• To provide a practical computational solution applicable to geographic and spatial analysis problems, such as finding poles of inaccessibility.
• To optimize the maximin distance from the circle center to all polygon edges, ensuring the largest possible radius.
• To deliver a method that is both computationally efficient and robust across a wide range of polygonal geometries.

Proposed method

• Uses distance geometry to model the minimum distance from a candidate center point to the polygon's boundary edges.
• Applies a gradient-based or iterative optimization technique to maximize the minimum distance (maximin criterion) to all edges.
• Employs a numerical approach that avoids the computational overhead of Monte Carlo sampling by focusing on critical boundary segments.
• Utilizes geometric properties of the polygon, such as edge directions and vertex positions, to refine candidate centers efficiently.
• Implements a convergence check based on distance tolerance to ensure stability and accuracy of the final center location.
• Supports both convex and non-convex polygons by dynamically adjusting the search space based on local curvature and edge proximity.

Experimental results

Research questions

• RQ1 How can the center of the largest inscribed circle in an irregular polygon be computed more efficiently than existing methods?
• RQ2 What optimization strategy minimizes computational cost while maintaining high accuracy in non-convex polygonal domains?
• RQ3 Can a deterministic geometric algorithm outperform stochastic methods like Monte Carlo in finding the pole of inaccessibility?
• RQ4 What are the key geometric and algorithmic components that ensure robustness across diverse polygon shapes?
• RQ5 How does the proposed algorithm compare in performance and precision to linear programming or heuristic-based approaches?

Key findings

• The proposed algorithm achieves faster convergence and higher precision compared to Monte Carlo-based methods for computing the largest inscribed circle.
• The method effectively handles non-convex and irregular polygons where traditional linear programming approaches may fail or be inefficient.
• The algorithm identifies the center of the largest inscribed circle by maximizing the minimum distance to polygon edges, ensuring optimal radius.
• The solution is robust across various polygon types, including those with complex or concave geometries.
• The computational efficiency is significantly improved over prior art, particularly in high-precision applications.
• The algorithm is applicable to real-world geographic problems, such as determining the pole of inaccessibility of landmasses.