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6] Daniel B. Karron. The \spiderweb" algorithm for surface construction in noisy volume data. The pattern of density readings created by the two curves in gure 11 are identical. Thus one cannot distinguish a double surface crossing (or any even number of crossings) between two adjacent sampling points from no crossing at all. The key point here is that there is no reason to interpolate an additional surface crossing on the bottom edge in each of the two faces shown in gure 11. The surface in face A is indistinguishable from the surface in face B. In fact the principal of Occam's Razor suggests that the linear interpolation used by SpiderWeb (edge hit rule) is perhaps the proper one. In analogy to the Nyquist Sampling Theorem for frequency sampled data, in order to detect such a curvature within a voxel unambiguously, one would have to sample the density at a larger number of points, i.e. increase the spatial frequency of the sampling. We can see in gure 12 that the minimum resolvable \U{turn" is such that the U-turn must straddle a voxel vertex (face A) while a U-turn completely contained within a voxel is unresolvable (face B). Thus if the minimal radius of curvature is greater than the width of a voxel, we will insure that the associated feature never lies entirely within a voxel. For the purposes of this discussion let us assume that the actual density has the following properties. First assume that at points where the function attains isovalues, the function does not achieve a local extremum. Secondly we assume that the set of isovalued points forms a two dimensional surface within the space. A simpler, but stronger, assumption is to require that the function be continuous in space. From these two assumptions we nd that any open ball containing a surface portion must also contain points of both higher and lower density than the isovalue. This means that the volume surrounding the surface contains both interior and exterior regions. Consider a single plane that properly slices the isosurface, so that there are no two dimensional regions of isovalues, as in gure 13. Note that following the above assumptions and the Jordan Curve theorem, the surface cannot trifurcate; in fact, it cannot split into any odd number of subsurfaces at any point in this plane. This would violate the division of this plane into interior and exterior …