Cross field singularity index violates Poincaré-Hopf theorem

[qnzhou]

Describe the bug

The singularity index computed via igl::find_cross_field_singularities is always positive, which leads to the violation of Poincaré-Hopf theorem (section 3.2 of [1]).

The original discussion and unit tests is from libigl/libigl-python-bindings#295

Platform

• Windows
• macOS
• Linux

[1] Vaxman, Amir, et al. "Directional field synthesis, design, and processing." Computer graphics forum. Vol. 35. No. 2. 2016.

[qnzhou]

Just to follow up, the function igl::find_cross_field_singularities seems to be unable to capture the singularities at the north/south pole of a sphere given the cross field following the longitude/latitude directions. This could be really problem leading to highly distorted seamless UV parameterizations.

[alecjacobson]

Oof. I don't personally use or maintain this part of libigl. Happy to merge a PR if it's known how to fix. Or even just a PR that documents the issue.