=-e-a--款 -}+如± (a) ce:(bi k=7+m+力T0+n立 女-顺 上s Se叫 val 1.F (b)isn hsopolage Form>15,al 1.the subdivision process is divergentsubdivision matrix. We index the eigenvalues of Sn in order of decreasing modulus as λi , i = 0, 1, . . . , 4n − 1. Note that four-sided faces are not always regular cases, since non-uniform Doo-Sabin surfaces permit arbitrary knot inter￾vals. For n = 4, the characteristic polynomial of S4 has the form C4(λ) = (λ − 1)(λ − 1 4 ) 5 (λ − 1 8 ) 4 (λ − 1 16 ) 4 ((λ − 1 2 ) 2 + (r0 + r2 − 1)(r1 + r3 − 1) 16 ). If (r0 + r2 − 1)(r1 + r3 − 1) ≤ 0, we get two real subdominant eigenvalues λ1 = 1 2 + √ −(r0 + r2 − 1)(r1 + r3 − 1) 4 , λ2 = 1 2 − √ −(r0 + r2 − 1)(r1 + r3 − 1) 4 . Since |(r0 + r2 − 1)(r1 + r3 − 1)| < 1, it follows that λ0 = 1 > λ1 > λ2 > λ3 = 1 4 . If (r0 + r2 − 1)(r1 + r3 − 1) > 0, one obtains two conjugate complex subdominant eigenvalues λ1 = 1 2 + √ (r0 + r2 − 1)(r1 + r3 − 1) 4 I, λ2 = 1 2 − √ (r0 + r2 − 1)(r1 + r3 − 1) 4 I. It is easy to know that λ0 = 1 > |λ1| = |λ2| > λ3 = 1 4 . Us￾ing the analysis techniques in [13], we prove that the proposed subdivision scheme is G 1 in both cases. For valence n = 3, similar results hold. Because the sub￾division matrix Sn has no obvious symmetries, it is difficult to perform an eigenanalysis for extraordinary points with valence n ≥ 5. Being analogous to Sederberg et al. [5], numerical ex￾periments and examples show that limit surfaces are G 1 at these points. 4.3. Examples Since NURDSes generalize biquadratic NURBS and Catmull-Clark-variant Doo-Sabin surfaces, a modeling pro￾gram based on NURDSes can handle any NURBS or Doo￾Sabin model as a special case. Fig. 11 shows uniform and non-uniform Doo-Sabin surfaces generated from a tetrahedron with holes. Fig. 11(a) depicts a uniform Catmull-Clark-variant Doo-Sabin surface after four refinement steps on the initial control mesh. Fig. 11(b) is an example of a NURDS in which three knot intervals along cer￾tain edges have been set to zero (as labeled), thereby creating a crease along the oval edge on the right. Fig. 12 is an example of a doughnut model whose initial con￾trol mesh is topologically a rectangular grid. Fig. 12(b) shows a uniform biquadratic B-spline surface. Fig. 12(c) is a biquadratic NURBS surface with a crease created by setting one row of the knot intervals to zero. Fig. 12(d) depicts a NURDS with a dart formed by setting to zero the knot intervals of appropriate ver￾tices. (a) (b) Figure 11: (a) A uniform Catmull-Clark-variant Doo-Sabin surface; (b) a NURDS with a crease. (a) (b) (c) (d) Figure 12: Doughnut model: (a) initial control mesh; (b) uniform biquadratic B-spline surface; (c) biquadratic NURBS surface with a crease; (d) NURDS with a dart. The shapes in Fig. 11(b) and Fig. 12(d) cannot be obtained using biquadratic NURBS or uniform Doo-Sabin surfaces. We next consider the configuration surrounding a type F face of valence n as illustrated in Fig. 10. Just like [11], we as￾sume that p0 = 1000 and all other knot intervals equal 1. For valence 3 ≤ n ≤ 30, we construct subdivision matrices for quadratic NURSSes and NURDSes respectively, and then in￾vestigate eigenstructure and continuity using the approaches de￾scribed in [13] with the help of a computer algebra system such as Mathematica. Fig. 13 plots absolute values of the first four eigenvalues of the subdivision matrix for quadratic NURSSes for 3 ≤ n ≤ 30. Concerning spectrum and continuity, we have the following results. • For 3 ≤ n ≤ 30, λ0, λ1 and λ2 may be negative, while λ3 is always positive. • For n ≥ 15, |λ0| > 1, the subdivision process is divergent. • For 3 ≤ n ≤ 14, λ0 = 1 > |λ1|, the subdivision process is convergent. 6