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Local Solid Shape

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ABSTRACT

Local solid shape applies to the surface curvature of small surface patches—essentially regions of approximately constant curvatures—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.

No MeSH data available.


A symmetrical saddle at various Casorati curvatures. The leftmost instance has curvature zero, thus is planar. For the planar case, the shape index is undefined; thus, this is not really a “saddle of zero Casorati curvature,” it is a mere “flat.” One might as well say it to be a “cap of zero Casorati curvature!” Thus, the planar case is not really a shape, just like “black” does not own a proper hue. In the real world “nothing” is the same as “anything” if you allow infinite discriminatory power! In the same sense, flat also means “no shape” as much as “any shape” (see also Koenderink et al., 2014).
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fig1-2041669515604063: A symmetrical saddle at various Casorati curvatures. The leftmost instance has curvature zero, thus is planar. For the planar case, the shape index is undefined; thus, this is not really a “saddle of zero Casorati curvature,” it is a mere “flat.” One might as well say it to be a “cap of zero Casorati curvature!” Thus, the planar case is not really a shape, just like “black” does not own a proper hue. In the real world “nothing” is the same as “anything” if you allow infinite discriminatory power! In the same sense, flat also means “no shape” as much as “any shape” (see also Koenderink et al., 2014).

Mentions: The magnitude should perhaps be defined such as to be 1/R for a sphere of radius R. The standard deviation is indeed such a measure (Koenderink & van Doorn, 1992)c=k12+k222the so-called Casorati curvature (Casorati, 1970). It is a non-negative magnitude of dimension [length]−1 that equals 1/R for a sphere of radius R. See Figure 1. The definition of the Casorati curvature as the standard deviation of the distances from the tangent plane is novel and will not be found in textbooks of differential geometry. It seems a particularly apt interpretation in the context of perception.Figure 1.


Local Solid Shape
A symmetrical saddle at various Casorati curvatures. The leftmost instance has curvature zero, thus is planar. For the planar case, the shape index is undefined; thus, this is not really a “saddle of zero Casorati curvature,” it is a mere “flat.” One might as well say it to be a “cap of zero Casorati curvature!” Thus, the planar case is not really a shape, just like “black” does not own a proper hue. In the real world “nothing” is the same as “anything” if you allow infinite discriminatory power! In the same sense, flat also means “no shape” as much as “any shape” (see also Koenderink et al., 2014).
© Copyright Policy - creative-commons
Related In: Results  -  Collection

License 1 - License 2 - License 3
Show All Figures
getmorefigures.php?uid=PMC5016822&req=5

fig1-2041669515604063: A symmetrical saddle at various Casorati curvatures. The leftmost instance has curvature zero, thus is planar. For the planar case, the shape index is undefined; thus, this is not really a “saddle of zero Casorati curvature,” it is a mere “flat.” One might as well say it to be a “cap of zero Casorati curvature!” Thus, the planar case is not really a shape, just like “black” does not own a proper hue. In the real world “nothing” is the same as “anything” if you allow infinite discriminatory power! In the same sense, flat also means “no shape” as much as “any shape” (see also Koenderink et al., 2014).
Mentions: The magnitude should perhaps be defined such as to be 1/R for a sphere of radius R. The standard deviation is indeed such a measure (Koenderink & van Doorn, 1992)c=k12+k222the so-called Casorati curvature (Casorati, 1970). It is a non-negative magnitude of dimension [length]−1 that equals 1/R for a sphere of radius R. See Figure 1. The definition of the Casorati curvature as the standard deviation of the distances from the tangent plane is novel and will not be found in textbooks of differential geometry. It seems a particularly apt interpretation in the context of perception.Figure 1.

View Article: PubMed Central - PubMed

ABSTRACT

Local solid shape applies to the surface curvature of small surface patches—essentially regions of approximately constant curvatures—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.

No MeSH data available.