Re: Need help with method of least squares

On Aug 4, 6:26 pm, toni.lass...@xxxxxxxxx wrote:

Actually they are the reciprocals of the eigenvalues of this matrix.
Therefore you need to compute the entire matrix.

Ok, so to find out the principal radii of curvature,

1. I find out the 2 X 2 matrix i.e. A00, A01, A10, A11.
2. I compute eigen values. The formula for the two eigen values is
given here :
http://mathworld.wolfram.com/Eigenvalue.html
3. I take reciprocals to find out the two principal radii of curvature


The author tries to approximate the triangulated surface locally
with a smooth surface that has a constant second fundamental form
over the entire triangle by taking finite differences over each of
the triangle edges. Of course, because you end up with more equations
than unknowns then such a surface does not usually exist. Instead he
takes a least-squares approximation to find a surface with constant
SFF that has almost the same curvatures as the triangular patch.


It's just basic linear least-squares.

Ok, I'm from computer science and I really don't understand this part.
Linear least squares require that you take the square of difference
between right and left hand sides of every equation and then sum it
up, and then find the minimum of the function. If there are two
unknowns x and y then you find the partial derivative of this function
with respect to x first to find value of x and then with respect to y
to find out the value of y. But in my case my unknowns are partial
derivatives themselves. So how to differentiate ??


Btw this gives the principal radii of curvature for a triangle. To
find principal radi of curvature at vertices, we simply take the
proc's of neighbouring vertices and calculate the proc of vertex using
some weighting factor for every triangle. Can I do the same for
edges ? i.e. if every edge in mesh is shared by two triangles, then
take contribution of two triangles.
.

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