Re: Need help with method of least squares

On Aug 4, 12:21 pm, pereges <Brol...@xxxxxxxxx> wrote:

Hello, I've been writing some program for calculating the principal
radii of curvature for a triangular mesh. I've been following this
thesis :

http://www.cs.princeton.edu/gfx/pubs/_2004_ECA/curvpaper.pdf

I got stuck up in page number 3, section 3.1 "Per face curvature
Computation"

He calculates a matrix lets call it A :

A00 = (del n / del u) dot u
A01 = (del n / del v) dot u
A10 = (del n / del u) dot v
A11 = (del n / del v) dot v

Now it seems he wants to calculate del n / del u and del n / del v
which give the principal radii of curvature.

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

There are three matrix equations given in the same page with a
triangle diagram. What I don't understand is why we want to solve
three equations when only two variables (del n / del u and del n / del
v) are needed.

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.

Further in the same page he says that they use method
of least squares to solve the equations. What is this method of least
squares in this context ?

It's just basic linear least-squares.
.

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