Subdivision Algorithms

H. Prautzsch, G. Umlauf

This research topic is part of the DFG-project "Subdivision Algorithms of arbitrary topology and smoothness", that began in 1997.

These algorithms are a popular tool in CAGD to generate arbitrary formed surfaces form control nets. Starting from an arbitrary initial net theses algorithms generate a sequence of ever finer nets that approximate the underlying surface better and better.

For the well known subdivision algorithm of Loop the differentiability of its limit surface was completely and rigorously proved for the first time. Proceeding on this proof a subdivision scheme was developed that generates curvature continuous surfaces. Thereby some free parameters can be used to optimise the fairness of the resulting surfaces.

Furthermore for a whole class of subdivision algorithms, the so-called mid-point schemes, the existence of a regular characteristic map without self-intersections was proved. Thus the existence of subdivision algorithms which generate arbitrarily smooth surfaces of arbitrary topology is established.