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Nov1-10, 12:34 PM
Hello. I was wondering if anyone here could help me understand this method I found described in the paper:
"A Practical Approach for 3D Model Indexing by combining Local and Global Invariants" by Jean-Philippe Vandeborre, Vincent Couillet and Mohamed Daoudi
for computing curvature indices for a polyhedron. I'm not very good at mathematics, so please try to dumb down your explanation accordingly, if possible ;)
-> To calculate the principal curvatures associated with a face (wording is mine, not an exact quote):
Fit a quadratic to the neighborhood of a face (the centroid of the face and those of its immediately adjacent faces) using the least square method;
Obtain the principal curvatures (two values, k1 and k2) as the eigenvalues of the Weingarten endomorphism W = I^-1 dot II where I and II are the first and second fundamental forms.
Now, I know what are centroids and eigenvalues, and how to calculate them. I do NOT know, and never used, least squares, Weingarten endomorphisms and fundamental forms, whatever they are. I cannot afford the time or cost of acquiring and learning from a book right now, but a quick wikipedia search returns this:
* Fundamental forms:
"the first fundamental form is the inner product on the tangent space of a surface"
"In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface"
This leaves me even more confused, and the formulas provided don't really match with my simple notion of inner product where (a,b).(c,d) = ac+bd ...
If anyone could explain this step in a simpler manner (even as a formula I can use in this specific instance), I'd be thankful. I might be able to figure it out if I had any idea of what I'm supposed to obtain as the end result, but I don't.
* Least squares:
"The minimum of the sum of squares is found by setting the gradient to zero. Since the model contains m parameters there are m gradient equations. (...)" etc.
This confuses me the most, since I have no idea of how to apply this to my set of centroids.
* Weingarten endomorphism:
I know it exists, but I can't find any encyclopedic information on it. However if I just have to use the provided formula, that won't be a problem, as long as I figure out the fundamental forms thing.
* Principal curvatures:
Wikipedia has a reasonably informative article on this, which is why I hope someone here will be able to help me. Unfortunately the article doesn't explain how to obtain the values.
"A Practical Approach for 3D Model Indexing by combining Local and Global Invariants" by Jean-Philippe Vandeborre, Vincent Couillet and Mohamed Daoudi
for computing curvature indices for a polyhedron. I'm not very good at mathematics, so please try to dumb down your explanation accordingly, if possible ;)
-> To calculate the principal curvatures associated with a face (wording is mine, not an exact quote):
Fit a quadratic to the neighborhood of a face (the centroid of the face and those of its immediately adjacent faces) using the least square method;
Obtain the principal curvatures (two values, k1 and k2) as the eigenvalues of the Weingarten endomorphism W = I^-1 dot II where I and II are the first and second fundamental forms.
Now, I know what are centroids and eigenvalues, and how to calculate them. I do NOT know, and never used, least squares, Weingarten endomorphisms and fundamental forms, whatever they are. I cannot afford the time or cost of acquiring and learning from a book right now, but a quick wikipedia search returns this:
* Fundamental forms:
"the first fundamental form is the inner product on the tangent space of a surface"
"In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface"
This leaves me even more confused, and the formulas provided don't really match with my simple notion of inner product where (a,b).(c,d) = ac+bd ...
If anyone could explain this step in a simpler manner (even as a formula I can use in this specific instance), I'd be thankful. I might be able to figure it out if I had any idea of what I'm supposed to obtain as the end result, but I don't.
* Least squares:
"The minimum of the sum of squares is found by setting the gradient to zero. Since the model contains m parameters there are m gradient equations. (...)" etc.
This confuses me the most, since I have no idea of how to apply this to my set of centroids.
* Weingarten endomorphism:
I know it exists, but I can't find any encyclopedic information on it. However if I just have to use the provided formula, that won't be a problem, as long as I figure out the fundamental forms thing.
* Principal curvatures:
Wikipedia has a reasonably informative article on this, which is why I hope someone here will be able to help me. Unfortunately the article doesn't explain how to obtain the values.