Study Of The Masks Of M-band Subdivision Algorithm Based On Triangulation Mesh

Subdivision algorithm has a long history, which dates to the metaphase and anaphase of 40's in the 20th century. Recently, subdivision algorithm has been put into computer graphics, computer aided geometric design(CAGD) and computer animation etc. M-band subdivision algorithm can resolve plenty of complex geometric shapes. In this paper, we discuss how to get the masks of M-band subdivision algorithm based on triangulation mesh.In chapter one, we introduce the main idea of subdivision and review the general cases of subdivision curves and surfaces in current years.In chapter two, some typical subdivision algorithms, such as Loop scheme, Catmull-Clark scheme, and Butterfly scheme are introduced. We also gave the geometric rules of these algorithms and the examples after subdivision.In chapter three, firstly, we introduce some correlative concept and property in the space of B-splines and the subdivision algorithm of the Box splines of uniform partition in three directions of S42(△). The expression of the M-band matrix of subdivision masks is given in this paper. Then, we approach the geometric rules of M-band subdivision and give the subdivision masks for the cases of M=4,5,6. At last, we prove all new vertex points, edge points and face points can be expressed to be a convex combination of no more than 12 old points.For spline function s(x,y), s(x,y) is rewritten over the refined grid asQ(x,y) is rewritten over the refined grid asFrom this,This is substituted into (1) to giveWe want to get the expression of ci,j above. Then, we can get the expression of the matrix of subdivision mask.According to affine transform, Fourier transform etc., it is found that, if we can compute the coefficients am,n in the following equation,we can get the expression of matrix A of subdivision mask.(Am,n= am-1,n-1,1≤m,n≤4M-3) Let M≥4 be integer, matrix A of subdivision mask can be expressed as1. When 0≤n≤m≤M-1, 2.When M≤m<2(M-1),m-M+2≤n≤M-1,3.When M≤m≤2(M-1),M≤n≤m,4.When 2(M-1)