CGAL users discussion list

[yanyajie <>]

Hi,

 

I try to locate ridge lines on point set surface. Now,

I have fitted an implicit surface F(x)=0, x=[x,y,z]’, to a point model (or, a point set).

According to Ohtake’s paper, “Ridge-valley lines on meshes via implicit surface fitting”,

one can obtain the normal n=n_x, n_y, n_z]’of a point on F using equation:

                                        n(x)=\nabla{F} / |\nabla{F(x)}|,

i.e., the normalized gradient vector. Then, the two non-zero

eigenvalues of 3x3 matrix \nabla{n(x)}, and the corresponding eigenvectors,

are the two principal curvatures and corresponding principal directions of x.

 

The results of normal computation are good. However, the eigen-decomposition of

the gradient of normal field gives me one real and two conjugate complex eigen-values

at many points. How should I deal with this situation? Does that mean these points

are umbilical points whose principal directions are undefined?

 

Has anyone read this paper and implemented the method described therein? Is my

understanding in terms of finding the principal curvatures correct? I have also looked

at the “approximating differential quantities” chapter in CGAL’s mannual, which estimates

the Weingarten Map. I think Weingarten Map is somehow equal to the gradient of normal

here. So, the method I am using here must make some sense.

 

Thanks in advance!

 

Yajie.