After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take O(N log N) operations to triangulate N points, although special degenerate cases exist where this goes up to O(N2).
The method entails deleting triangles whose circumcircles enclose the newly inserted point, yielding a star-shaped polygonal gap that is then re-triangulated with the new point. By utilizing the connectivity of the triangulation, the algorithm can efficiently locate and remove triangles in O(N log N) time, although special cases may result in O(N2).