Local Solid Shape
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fig2-2041669515604063: A cap, ridge, saddle, rut, and cup. The shape indices are, respectively, −π/2, −π/4, 0, +π/4, +π/2. These have the same Casorati curvature. The colors are those of the hue scale used in this article. Notice that complementary colors denote complementary—related as mold and cast—shapes. Thus, the cap fits the cup, the ridge the rut, whereas the saddle fits itself. The scale is actually continuous, thus—for example—there is a smooth range between cap and ridge. The symmetrical caps and cups are special, technically known as “umbilical.” Likewise, the saddle shown here is special, the “symmetrical saddle” or “minimal surface.” The ridge and ruts are mere transition points. However, when coarse graining, they also subtend finite (though fuzzy) ranges.
Mentions: A quality measure is more involved. It should evidently be a dimensionless number. However, the ratio has the drawback that it is invariant with respect to a simultaneous sign change of both principal curvatures. Such an operation inverts the z-coordinate, thus it transforms a form into its negative, related to the original like the mold to the cast. An example is the inside and outside of an egg shell. Notice that the symmetrical saddle is congruent to its own mold! Thus, it would be natural that it had measure zero (because +0 equals −0). Another useful observation is that the case of equal principal curvatures is very special, and you cannot get any “rounder” than that. Thus, a quality measure should be defined on a finite, symmetrical segment like [−α,+α] (with constant α > 0), where zero corresponds to the symmetric saddle and the endpoints to the inside and outside of spherical shells. An example of such a measure is (Koenderink & van Doorn, 1992)s=arctan(k2+k1k2-k1)which denotes the shape index. It evidently varies on the segment [−π/2,+π/2]. See Figure 2.Figure 2.

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ABSTRACT

Local solid shape applies to the surface curvature of small surface patches&mdash;essentially regions of approximately constant curvatures&mdash;of volumetric objects that are smooth volumetric regions in Euclidean 3-space. This should be distinguished from local shape in pictorial space. The difference is categorical. Although local solid shape has naturally been explored in haptics, results in vision are not forthcoming. We describe a simple experiment in which observers judge shape quality and magnitude of cinematographic presentations. Without prior training, observers readily use continuous shape index and Casorati curvature scales with reasonable resolution.

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