Automatic Construction of Polyhedral Surfaces From Voxel Representations

Cover Automatic Construction of Polyhedral Surfaces From Voxel Representations
Automatic Construction of Polyhedral Surfaces From Voxel Representations
Alan Shaw
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We call this a "birth" event. At this level the section consists of a single point, which immediately opens into a loop as we ascend. A series of loops is produced until a second critical pxjint of the doughnut is reached. At this point, the loop becomes a "figure eight", and splits into two disjoint loops. We call this event "splitting. " Then, at the third critical point, the loops fuse back into a "figure eight". We call this a "merge" event. The figure eight immediately becomes a simple loo...p. Finally, the single loops shrink down to a point and disappear at the fourth and final criti- cal point (a "death" event).
For the embedding of the torus implied by this sectioning, there are four isolated critical points. It can be proven that any differentiable 2-manifold can be embedded in Euclidan 3-space in such a way that its critical points are finite in number, and no more than one occurs on a given level. Also, it can be shown that the only topological changes which can occur are the "birth", "death", "splitting" and "merging" events illus- trated for the torus ^ Each topological change is accompanied by an Euler transition, that is, a change in the Euler number of a contour.


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