External Incremental Delaunay Triangulation

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.