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eigendecomposition of shape operator #18

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aPToul opened this issue Mar 20, 2018 · 1 comment
Open

eigendecomposition of shape operator #18

aPToul opened this issue Mar 20, 2018 · 1 comment

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@aPToul
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aPToul commented Mar 20, 2018

During principal curvature estimation (via the shape operator), I am having some trouble getting real values.

The EigenSolver for eigendecomposition finds complex values / vectors, but from the argument in the ReadMe, the shape operator is diagonalizable with real values, since curvature is real and extremal.

I'm wondering if there's a way to encourage the solver to find the closest real valued decomposition (I haven't had a chance to read up on the fundamental forms to verify whether they are indeed always decomposable this way, but I guess it should be the case for the quadratic form).

@aPToul
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aPToul commented Mar 20, 2018

Looks like Eigen::SelfAdjointEigenSolverEigen::Matrix2d PCA_2D could be promising,
but it assumes our operator is self adjoint. This is fine since the product of two symmetric matrices (L * R) is symmetric, and the inverse of a symmetric matrix (L) is symmetric. I think this will work.

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