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Extra resources for ACM Transactions on Graphics (January)
Fig. 11. Characteristic maps for several different configurations of quads and triangles. According to Levin and Levin , if there exists a k such that ρ [k] < 14 , then the subdivision scheme is C 2 at the boundary. The obvious choice for constructing the matrix W is to simply use all of the eigenvectors of A. However, this approach can be numerically unstable if the matrix has small eigenvalues. Levin and Levin  suggest that W be formed from the right eigenvectors associated with the eigenvalues from and a basis of the null space from the corresponding left eigenvectors.
10. Photograph of an auroral spiral. ) Fig. 11. Simulation of an auroral spiral showing a rayed structure and an arc break up. Fig. 12. Frames of a simulation sequence showing the motions and shape changes of an auroral rayed band. ACM Transactions on Graphics, Vol. 24, No. 1, January 2005. 52 • G. V. G. Baranoski et al. Fig. 13. Black and white all-sky photographs showing different stages of an auroral surge formation. ) simulations were 2000km, and 9km, respectively. The computations were performed on a SGI550, with dual 850Mhz Pentium3 processors.
Composed of triangles and quads subdivided using our method. On the bottom of the figure, we extracted the cerrebellum and subdivided the model with the triangle/quad structure explicitly shown. 3. 1 ANALYSIS Necessary Conditions Given an ordinary triangle/quad boundary (shown in Figure 8), we define S to be the subdivision matrix for Stam and Loop’s  scheme, formed by centroid averaging. For the subdivision scheme to be C 2 in the functional sense, S must satisfy Sz i = λi z i (1) where λi = 1, 12 , 12 , 14 , 14 , 14 and z i are the corresponding eigenvectors producing the polynomials 1, x, y, x 2 , xy, y 2 [Warren and Weimer 2001].