Deterministic Linear Time Constrained Triangulation using Simplified Earcut

TL;DR

This paper introduces a simple, deterministic linear-time algorithm for triangulating polygons with constraints, improving efficiency and simplicity over previous methods by leveraging properties of convex vertices and ears.

Contribution

The paper presents a novel, simplified earcut-based triangulation algorithm that guarantees linear time complexity for constrained polygons, with formal correctness proof and practical validation.

Findings

Algorithm operates in linear time for constrained triangulation.

Formal proof confirms correctness and convergence.

Practical experiments show efficiency gains over prior methods.

Abstract

Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to: (1) deleting all the triangles it intersects; (2) filling the so generated hole with two polygons that have the wanted segment as shared edge; (3) triangulate each polygon separately. In this paper we prove that these polygons are such that all their convex vertices but two can be used to form triangles in an earcut fashion, without the need to check whether other polygon points are located within each ear. The fact that any simple polygon contains at least three convex vertices guarantees the existence of a valid ear to cut, ensuring convergence. Not only this translates to an optimal deterministic linear time triangulation algorithm, but such algorithm is also trivial to implement. We formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art.