The curvature is zero for a point in the interior of a face and isQuote:> How to calculate a curvature of 3D polyhedron?
> Is there some survey/article ?
Probably what you are looking for is to approximate the polyhedron by a
C2-surface, then compute the principal curvatures for that surface. The
typical approach is to estimate first and second partial derivatives at a
vertex of the polyhedron by using finite differences related to all the
vertices connected to the given vertex by edges.
>The curvature is zero for a point in the interior of a face and is
>discontinous for a point on an edge or at a vertex.
Sorry, Joe, but I do not know what you mean by a "strict definition".Quote:> Just to supplement what Dave says above, the curvature is zero everywhere
> except at the vertices. Even along an edge the curvature is zero.
> (This is under a strict definition of curvature; you may want instead
> the curvature of a smooth approximation of the polyhedron.)
The classical construction for *a* curvature is the following. Given a
point P on a surface, suppose a normal vector to the surface at P is
N. Select a tangent vector T at P. The plane containing P and spanned
by N and T locally intersects the surface in a curve that can be
parameterized by Y(t) where Y(0) = P. The curvature for that curve at
t = 0 is constructed in the usual way for a planar curve. Since there are
infinitely many tangent vectors to select, there are (usually) infinitely
many curvatures you can measure at P. To ask the question "what is
the curvature of the surface at P" is not well-posed. The metric and
curvature tensors contain all the information you need to construct the
curvature associated with a tangent T. Principal curvatures are the
two extreme values of curvature as you vary over all T. If the curvature
is constant for all T at P, then you have an umbilic point and the surface
is locally spherical.
This construction relies on the fact that you actually have a normal N
and a tangent space at P. Any point on the shared edge between
polyhedral faces does not have a normal vector, so I fail to see how
you can apply the standard construction above. Even if you were to
select some N and call it a normal, you only get one T for which the
curvature at P within that slice is defined (T is the edge direction,
curvature is zero). But for any other T, you get a cusp in the curve
of intersection and the curvature is undefined.
>Sorry, Joe, but I do not know what you mean by a "strict definition".
>The curvature tensor for a surface parameterized by X(s,t) involves
>second derivatives in s and t. [...]
>[...]To ask the question "what is
>the curvature of the surface at P" is not well-posed.
Thanks for the clarification of your definitions.Quote:> There are so many notions of curvature that perhaps we are not in
I was discussing the curvature tensor in general. The intrinsicQuote:> The "intrinsic," "Gaussian" curvature is indeed defined
> at a point, at any point p of a continuous surface. It can be defined
> without mentioning derivatives, as the limit of a ratio of the angle
> deficit of a geodesic polygon enclosing p to the area of that polygon.
> I believe you are discussing the "extrinsic" curvature, which depends
> on its embedding in space.
For an interior point on an edge of a polyhedron, the intrinsic
curvature (Gaussian curvature) is zero and the extrinsic
surface is the dihedral angle between the faces sharing the edge.
A vertex does not have an intrinsic curvature, but it does have
an extrinsic curvature given by the solid angle formed by the
normals of the faces that share that vertex.
You are not muddying the waters since it is not clear what theQuote:> No doubt the original poster was not looking for Gaussian
> curvature, and I apologize for muddying the waters.
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