Computing the minimal perimeter polygon for digital objects in the triangular tiling

2026-09-12

AI Summary

本文提出了一种算法来计算三角平铺中数字对象的最小周长多边形，通过构建和迭代约束可见性锥体以及追踪边界路径实现。

Abstract

This work presents an algorithm, together with its correctness proof, to determine the minimum perimeter polygon (MPP) for digital objects given as regular complexes in the triangular plane tiling. Such objects are edge-adjacency-connected sets of triangle tiles that have no end tiles, and the point set union of all their tiles forms a simple polygon. Nevertheless, the boundary paths of the objects are not assumed to be simple. Then the MPP is a weakly simple polygon that coincides with the relative convex hull (i.e., geodesic hull) of a set $A$ with respect to a simple polygon $B$, where $A\subset B$, but $A$ is not necessarily a polygon, in fact it is generally not connected. Our MPP algorithm relies on constructing and iteratively constraining cones of visibility through forthcoming boundary tiles, it uses the structure of the canonical boundary path, the MPP frontier is the shortest polygonal curve following this path. We also propose a boundary tracing algorithm to obtain such paths from the objects.

Problem

• minimum perimeter polygon
• triangular tiling
• digital objects
• relative convex hull
• geodesic hull

Innovation

• minimum perimeter polygon
• triangular tiling
• visibility cones
• canonical boundary path
• boundary tracing algorithm