errata: of the version in the book "Fair curves...", corrected in the e-print.

Note that the reparametrization functions psi and phi are correct in either case and any formulas in doubt can be derived from them.
p 6, l2 should be
P_{11,i} = {1\over 36}(5 B_{i,2}+ 5 B_{i,1}+ A_i+ 25C_i) + {a\over 9}( C_i- A_i)
P_{21,i} = {1\over 36}(5 B_{i,2}+ 5C_{i+1}+ B_{i+1,1}+ 25C_i) \cr
&\quad + {a\over 18}( -5 B_{i,2} + B_{i+1,1} + C_{i+1} + 3C_i)
E_i := (1 - cc) P_{32,i} + cc P_{31,i} cc = 2c/3
p 8, Corollary should be: v-c pair
p 10, (I_4) should be:
2( (1-cc) P_{32,i} + cc P_{31,i}) = P_{22, i+1} + P_{22,i} ( i+1 to be consistent with the counterclockwise orientation of Fig 1.2.2)

Answer The correspondance mesh cell to patch is actually natural (the way you would expect each cell to be covered by several patches). The figure to look at is the one that shows the smoothed cube, right in the beginning of the paper. You want to notice that the Doo-Sabin split creates a dual mesh for the intermediate control points C_i which themselves are dual to the quadratic patches. So dual(dual) == primal and the patch boundaries are parallel to the cell boundaries.

Question However, I have now some problems to see, that the construction in Sapidis book always is C^1. What happens if I have a 6-sided cell in the original control mesh and one of the 6 vertices of this cell has a degree 8 for instance? Do the perturbations, you are describing, always guarantee C^1 continuity in this case? Answer The construction is always $C^1$ and your question is addressed by Corollary 11.3.2. Look at Figure 11.3: The quantities to be perturbed are B_{i,j}. They lie in quadrilaterals that are edge-adjacent to the irregular-vertex or irregular-face panel. Now draw a picture of two refinements; you will see there is no overlap between edge-adjacent quadrilaterals belonging to irregular-vertex or irregular-face panels. (That is why this method needs 2 refinements! 9395ffss uses just one refinement and can therefore not in general perturb)

in German Die Gr"o3e a (auf Seite 6 des online papers) ist a = c/(1-c) was f"ur n-> infty und daher c->1 recht gro3 werden kann. Hier sind Beispiel "a"-Werte:
3 = -.3333333333, 4 = 0, 5 = .4472135958, 6 = 1., 7 = 1.655970555, 8 = 2.414213560, 9 = 3.274316085, 10 = 4.236067981
Im Gegensatz zu den C1-surface splines ist es hier wohl nicht m"oglich die konvexe Huellen Eigenschaft zu zeigen (dazu musste man mehr "uber A, B und C wissen.) Deswegen steht in der homepage: For general modeling I recommend C^1-surface splines. Surfaces...bicubics is older, uses more patches, uses more complicated formulas, does not prove the convex hull property. It is however the first of this type of construction.

Question (in German) Diese neuere Arbeit produziert in gewissen Faellen nur approximativ C1, wenn ich mich richtig erinnere. Deshalb habe ich mich auf die andere Arbeit verlegt.

Answer (in German) Fur bikubische patches, wenn man eine 2te Unterteilung wie im Falle des Sapidis papers erlaubt und genau wie dort eine perturbation macht, dann ist die Fl"ache immer C1 (das is die Bemerkung in Q/A) Die kubischen (dreieck-patch) Fl"achen sind immer C1.