[vtkusers] vtkCurvatures: mean curvature

[jarv0075]

I believe this mean curvature implementation is a bit different than the
one used in VTK (since there's no certified -right- way of going from
continuous to discrete curvature), but the following paper:

   http://www-grail.usc.edu/pubs/DMSB_III.pdf

should help explain why it is how it is.  (Took me a while too).

HTH,



-Tim

---------------------------------------
 Timothy R. Jarvis                   
---------------------------------------
Graduate Research Assistant 
International Neuroimaging Consortium
VA Medical Center
612-467-2619
http://www.neurovia.umn.edu
---------------------------------------

On 28 Sep 2004, Bernard Chiu wrote:
> Dear vtk users,
> 
> I am a novice in differential geometry and I would like to ask the
experts
> about a number of issues in the implementation of the class vtkCurvature:
> 
> (1)I understand that the mean curvature of an vertex was calculated by
> averaging the curvature along each edge connected to the vertex. However,
I
> don't understand how the curvature along an edge can be calculated by
H(edge
> e) = length(e)*dihedral_angle(e). In addition, curvature calculated using
> this equation seems to be scaled by a factor (3/Af) in the code, where Af
is
> the total area of two neighbouring triangles that have edge e as their
> common edge.
> Could anyone explain why the curvature of each edge was computed this
way?
> 
> (2) In the calculation of the mean curvature, the comments state that the
> normal was assumed to be pointing outwards. In the code, the vertices of
a
> triangle were retrieved in an order according to its id. I thought the id
of
> a triangle can be ordered arbitrarily, and therefore, I don't understand
how
> we can be sure that the normal always points outwards.
> 
> (3)Since the order of the vertices in a triangle is arbitrary, there are
two
> choices in the direction of the unit vector pointing along the direction
of
> an edge (this unit vector is denoted by e in the code) (one of the
choices
> has direction exactly opposite to each other). The way in which the sine
of
> the dihedral_angle was calculated would imply that the dihedral_angle is
not
> uniquely defined, even if the normal of the triangles are defined as
> pointing outwards. Then, would it mean the curvature itself is not
uniquely
> defined?
> 
> Thanks in advance for your help,
> 
> Bernard Chiu
> 
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