6.1 A hypersurface construction algorithm.

The design of SpiderWeb and its proof of correctness follow the proof of theorem 1.

For a given threshold $\tau$ and for a given 3-dimensional cube, all vertices are labeled High or Low as above. A hit point (hit for short) is interpolated on the interior of every cube edge that has a High and Low vertex. We now define hit adjacency, as in the proof of theorem 1.

Definition 24   In a 4-hit cube face, if $\tau$ is above the disambiguation value, a pair of hits on two face edges sharing a High vertex are adjacent (knit High). For $\tau$ below the disambiguation value, a pair of hits on two face edges sharing a Low vertex are adjacent (knit Low). Two hits are adjacent if they are on the same 2 hit cube face, or if they are adjacent on a 4-hit face. We define connectivity to be the transitive closure of the adjacency relation.

A connected component of hits within a hypercube corresponds to a hypersurface patch of $B (\bf {X}_{f}^{\geq \tau})$.

The algorithm is illustrated in figures 25 through 28, and original references can be found in [14] [15] [16] [17] [18] [5].