Z.Hung ct al./of Information Compurtational Science 7:1 (20110)1-6 3 Extended Doo-Sabin Surfaces 3.1 Regular faces d P。d Fig.3:A regular face with its knot intervals F-P+V +Ba-Pi-Pe3) *-学P,+学+学P+学+学P+学P e taken modulo Let the c4 Z. Huang et al. /Journal of Information & Computational Science 7: 1 (2010) 1–6 n > 12. On the other hand, for convergent quadratic NURSSes, no closed-form limit point or limit normal rules have been proposed so far. 3 Extended Doo-Sabin Surfaces We now present a modification of quadratic NURSSes [3] that we call EDSes (Extended Doo￾Sabin Surfaces). EDSes are identical to quadratic NURSSes, with one difference: EDSes use the refinement rules for Doo-Sabin surfaces [1] to compute new vertices created for irregular faces. It is for that reason that EDSes are always convergent for all positive knot intervals. It is also for that reason that EDSes have closed-form limit point as well as limit normal rules. 3.1 Regular faces bc bc bc bc P0 d01 d03 P1 d12 d10 P2 d23 d21 d30 P3 d32 b b b P0 b P1 P2 P3 Fig. 3: A regular face with its knot intervals. The refinement rules for regular faces for EDSes are identical to those for quadratic NURSSes. Let n = 4, it follows from Eq. (1) that Pi = Pi + V 2 + ci+2 4 P3 k=0 ck (Pi−1 + Pi+1 − Pi − Pi+2) = (1 2 + wi 2 − wi+2 4 )Pi + (wi−1 2 + wi+2 4 )Pi−1 + (wi+1 2 + wi+2 4 )Pi+1 + wi+2 4 Pi+2 where wj = cj/ Pk=3 k=0 ck, j = 0, 1, 2, 3, and indices are taken modulo 4. We now derive limit point and limit normal rules for regular faces. Let the control point vector of a regular face be M = [P0, P1, P2, P3] T and M be the corresponding control point vector after