Necessary Conditions for Subdivision Surfaces

Subdivision surfaces are considered which consist of tri- or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme. Keywords Subdivision, piecewise polynomial surface, arbitrary topology, extraordinary point. 1 Introduction This paper deals with subdivision algorithms acting on arbitrary 2D control nets as the schemes described in [Doo78, DS78, CC78, Loo87, Qu90, DGL90, Kob94, PR96]. Here we will generalize the degree estimate in [Rei94]. A surface S generated by one of the above mentioned algorithms consists of infinitely many either tri- or quadrilateral patches which are locally arranged as illustrated in Figure 1 schematically. Note that these patches may consist of smaller patches themselves as indicated by the broken lines. For our analysis it suffices to consider such a local patch configur..