Successive Overrelaxation–Progressive Interpolation for Loop Subdivision Surfaces

The Loop subdivision scheme is one of the most widely used approximation subdivision methods for generating smooth surfaces. However, the limiting surface of the Loop subdivision scheme does not interpolate the vertices of the original mesh and may exhibit shrinkage in certain cases. To overcome this limitation, we propose the Successive Overrelaxation–Progressive Iterative Approximation (SOR-PIA) method, which adjusts the positions of the original vertices so that the corresponding limit surface of the Loop subdivision scheme can interpolate the vertices of the given mesh. The optimal relaxation parameter for the SOR-PIA method is also provided. Compared to classical PIA, Weighted PIA (W-PIA), Hermitian and skew-Hermitian PIA (HSS-PIA), and Weighted Hermitian and skew-Hermitian PIA (WHSS-PIA), the proposed method converges faster while maintaining accuracy, as demonstrated by numerical examples.