External Incremental Delaunay Triangulation
Yifeng Cai
TL;DR
This work presents an external incremental method for 2D Delaunay triangulation that uses the convex hull as the core operation unit, avoiding the triangle containment search inherent in traditional triangle-based incremental methods. Points are processed in coordinate-sorted order, and each new point is integrated by finding the upper and lower tangents to the current hull, deleting edges whose circumcircles contain the new point, and then linking exposed hull points to form a new triangulation; the algorithm relies on data structures P, L_ch, and D_tri to manage points, hull, and triangles. Key contributions include a convex-hull–based incremental construction with $O(n log n)$ point sorting, $O(n)$ tangent searches, and empirical linear-time performance for uniform point sets, along with a correctness proof grounded in Voronoi duality and edge flips. The approach simplifies implementation by removing the need for a fictitious outer triangle and dynamic triangle search structures, potentially yielding practical performance benefits for large, well-distributed point sets.
Abstract
This paper introduces a Delaunay triangulation algorithm based on the external incremental method. Unlike traditional random incremental methods, this approach uses convex hull and points as basic operational units instead of triangles. Since each newly added point is outside the convex hull, there is no need to search for which triangle contains the point, simplifying the algorithm implementation. The time complexity for point sorting is $O(n\log n)$, while the collective complexity for upper/lower tangent searches is proven to be $O(n)$. For uniformly distributed point sets, empirical results demonstrate linear time $O(n)$ for full triangulation construction. The overall time complexity remains $O(n\log n)$. This paper details the algorithm's data structures, implementation details, correctness proof, and comparison with other methods.