Hello!

How to calculate a curvature of 3D polyhedron?

Is there some survey/article ?

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 ?

discontinous for a point on an edge or at a vertex.

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.

--

Dave Eberly

http://www.magic-software.com

>> How to calculate a curvature of 3D polyhedron?

>> Is there some survey/article ?

>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.

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.) One can easily

compute the curvature concentrated at each vertex as 2 pi minus the sum of

the incident face angles. Thus the curvature at each corner of a cube

is pi/2. The Gauss-Bonnet theorem says that the curvature summed over

the whole surface, i.e., the sum over all vertices of a polyhedron,

is 4 pi. For the cube: 8 * (pi/2) = 4 pi.

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 curvature tensor for a surface parameterized by X(s,t) involves

second derivatives in s and t. Parameterize two faces of a

polyhedron that share an edge. Neither the first nor second derivatives

exist along the edge, so you cannot compute the curvature tensor.

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.

--

Dave Eberly

http://www.magic-software.com

>> 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.)

>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.

disagreement. 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 a point p on the interior of a polyhedron edge, a little

geodesic polygon (edges composed of geodesics) around p flattens

to just a polygon. Its total angle turn is 2pi, so its angle deficit

is 0. So the limit of its deficit to area is zero. So the Gaussian

curvature at p is zero.

No doubt the original poster was not looking for Gaussian

curvature, and I apologize for muddying the waters.

Thanks for the clarification of your definitions.Quote:> There are so many notions of curvature that perhaps we are not in

> disagreement.

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.

curvature on a smooth surface is the Gaussian curvature. The

extrinsic curvature is the mean curvature. I had to dig out my

"Solid Shape" book by Jan Kooenderink (it was a bit dusty).

Just for reference for future searches:

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.

original poster wanted. Perhaps he/she will clarify.

--

Dave Eberly

http://www.magic-software.com

1. ANN: BspSolid -- Robust and efficient Polyhedral Solid Modeling

Hi All,

I am very pleased to announce the availability of robust and efficient

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Once you are there, go to "Downloads" to download shareware programs,

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Thanks and Enjoy !

Dennis Jiang

http://www.ncf.carleton.ca/~ds755/

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