Subdivision Depth Computation for Tensor Product n-Ary Volumetric Models

and Applied Analysis 3 P 1 2i P 1 2i 1 P 1 2i 2 P i 1 P 1 2i 3 P 1 2i 4 P k i 2 P k i a P 1 3i P 1 3i 1 P 1 3i 2 P i 1 P 1 3i 3 P 1 3i 4 P 1 3i 5 P 1 3i 6 P k i 2 P k i b P 1 4i P 1 4i 1 P 1 4i 2 P k i 1 P 1 4i 3 P 1 4i 4 P 1 4i 5 P 1 4i 6 P 1 4i 7 P 1 4i 8 P k i 2 P k i c Figure 1: Solid lines show coarse polygons whereas doted lines are refined polygons. a – c represent binary, ternary, and quaternary refinement of coarse polygon of scheme 2.1 for n 2, 3, 4, respectively. 2. Preliminaries In this section, first we list all the basic facts about subdivision curve, surface and volumetric models needed in this paper. Then we settle some notations for fair reading and better understanding of Section 3. 2.1. Concepts 2.1.1. n-Ary Subdivision Curve Given a sequence of control points p i ∈ N , i ∈ , N 2, where the upper index k 0 indicates the subdivision level, an n-ary subdivision curve 5 is defined by p 1 ni α m ∑ j 0 aα,jp k i j , α 0, 1, . . . , n − 1, 2.1 where m > 0 and m ∑ j 0 aα,j 1, α 0, 1, . . . , n − 1. 2.2 The set of coefficients {aα,j , α 0, 1, . . . , n − 1}mj 0 is called subdivision mask. Given initial values p0 i ∈ N , i ∈ , then in the limit k → ∞, the process 2.1 defines an infinite set of points in N . The sequence of control points {p i } is related, in a natural way, with the dyadic mesh points tki i/n k , i ∈ . The process then defines a scheme whereby p 1 ni α replaces the value p i α/n for α ∈ {0, n}. Here p k 1 ni α is inserted at the mesh point t k 1 ni α 1/n n−α t k i αt k i 1 for α 0, 1, . . . , n. Labelling of old and new points is shown in Figure 1 which illustrates subdivision scheme 2.1 . 2.1.2. Tensor Product n-Ary Subdivision Surface Given a sequence of control points p i,j ∈ N , i, j ∈ , N 2, where the upper index k 0 indicates the subdivision level, a tensor product n-ary surface is a tensor product of 2.1 defined by p 1 ni α,nj β m ∑ r 0 m ∑ s 0 aα,raβ,sp k i r,j s, α, β 0, 1, . . . , n − 1, 2.3 4 Abstract and Applied Analysis P 1 2i,2j 2