Conclusion

The analysis of stationary linear subdivision algorithms presented in this book summarizes and enhances the results of three decades of intense research and it combines them into a full framework. While our understanding of C¹-algorithms is now almost complete, the generation and the analysis of algorithms of higher regularity still offers some challenges. Guided subdivision and the PTER-framework pave a path towards algorithms of higher regularity, that, for a long time, were considered not constructible. Various aspects of these new ideas have to be investigated, and the development is in full swing, at the time of writing.

In focusing on analytical aspects of subdivision surfaces from a differential geometric point of view, we left out a number of other interesting and important topics, such as the following.

• Implementation issues: In many applications, subdivision is considered a recipe for mesh refinement, rather than a recursion for generating sequences of rings. The availability of simple, efficient strategies for implementation [ZSOO, SAUK04], even in the confines of the Graphics Processing Unit [BS02, SP03, SJP05a], evaluation of refinable functions at arbitrary rational parameters [CDM91] and, for polynomial subdivision, at arbitrary parameters [Sta98a, Sta98c] and their inclusion into the graphics pipeline [DeR98, DKT98] largely account for the overwhelming success of subdivision in Computer Graphics.

• Sharp(er) features: By using modified weights for specially ‘tagged’ vertices or edges, it is possible to blend subdivision of space curves and subdivision of surfaces to deliberately sharpen features and even reduce the smoothness to represent creases or cusps [DKT98,Sch96]. In a similar way, subdivision algorithms can be adapted to match curves and boundaries [Nas91,Lev99c,Lev00,Nas03].

• Multiresolution: Based on the inherent hierarchy of finer and finer spaces, one can develop strategies for multiresolution editing of subdivision surfaces [LDW97]. There are close relations to the study of wavelets, but this development is still in its infancy.

• Applications in scientific computing: Beyond the world of Computer Graphics, subdivision surfaces can be employed for the simulation of thin shells and plates [COSOO, GKS02, Gri03, GTS02], and possibly also in the boundary element method.

Many of these and other application-oriented issues are discussed in the book of Warren and Weimer [WW02].

Necessarily, the material presented here is a compromise between generality and specificity. Therefore, to conclude, we want to review the basic assumptions of our analysis framework, check applicability to the rich ‘zoo’ of subdivision algorithms in current use, and discuss possible generalizations. We consider in turn function spaces, types of recursion, and the underlying combinatorial structure.