Bounding the distance between the kth control mesh and the limit surface of Li’s subdivision

Abstract Subdivision schemes generate smooth surfaces by iteratively refining a coarse initial mesh. However, it is important to determine how many iterations are needed to achieve the desired subdivision surface while adhering to the pre-defined error tolerance, which is known as the subdivision depth. We introduce an advanced technique to compute the subdivision depths of Li’s scheme presented in (Comput. Graph. Forum 24(1), 3–16, 2005). This technique involves calculating the distance between consecutive levels of meshes, and then calculating the distance between any arbitrary level of mesh and the limiting surface, which is known as error bounds. Then correlate the mask of the scheme with an arbitrary vector and obtain expressions through convolution. As the order of the convolution increases, the values of these expressions monotonically decline. As a result, one can obtain sharper error bounds and fewer values for subdivision depths by increasing the order of convolution. The convolution allows for balancing the number of iterations and the predefined errors, making the process more efficient. The algorithms are introduced to make things easier and more understandable for the readers when using applications. The validity of the algorithms are confirmed by testing it on different parts of the mesh with varying valence numbers. The versatility of the proposed approach is demonstrated through the creation of tables and graphs.