Next: Evaluation and Derivative Masks Up: Reconstruction of Limit Surface Previous: Reconstruction of Limit Surface


Recursive Interpolating Subdivision

The first smooth interpolating scheme was introduced in [20] which is known as the butterfly scheme. Like all interpolating schemes, each step of the butterfly subdivision scheme leaves the existing nodes unmoved and uses the local averaging (Fig. 46) to compute the position of midside nodes introduced during the splitting. However, the butterfly scheme exhibits degeneracies when applied to grids of arbitrary topology, which makes its use quite limited. Therefore, in the presented approach, the recursive subdivision based on the modified butterfly scheme [72] is employed. This is the interpolating non-uniform stationary scheme, in which the position of existing nodes (on the current level of the subdivision) remains unchanged and the position of a new node on the next level (Fig. 48) is calculated as

(90)


where are nodes connected to the surface node of valence . The weights and corresponding to the surface averaging mask (Fig. 48) are given by

(91)
(92)
(93)
(94)

The application of the modified butterfly scheme to a star-shaped polyhedron is presented in Figure 50. The geometric similarity of the refined grid with the initial control grid is maintained through the whole process of the subdivision up to the limit surface.

The modified butterfly scheme exhibits favourable properties which make the scheme very powerful and which can be identified as:

Similarly, the limit boundary curves are recovered using a one-dimensional interpolating subdivision [10] producing continuous curves. The adopted 4-point (for a new node between two curve nodes) and 3-point (for a new node between vertex node and curve node) averaging masks are depicted in Figures 49a and 49b.

The final interpolating procedure evaluates the position of a new node according to the classification and regularity of the end nodes of its parent edge:



Next: Evaluation and Derivative Masks Up: Reconstruction of Limit Surface Previous: Reconstruction of Limit Surface

Daniel Rypl
2005-12-07