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).
After each insertion, any triangles with circumcircles encompassing the new point are eliminated, leaving behind a star-shaped polygonal cavity that is then re-triangulated utilizing the new point. By exploiting the connectivity of the triangulation, the algorithm can accomplish O(N log N) operations to triangulate N points, although exceptional degenerate cases may lead to O(N2).