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The Wikipedia article on umbilical points classifies umbilical singularities into three types ─ star, monstar, and lemon ─ depending on the index of the singularity and the configuration of the lines of curvature around the singularity,

and it also provides a nicely visualizable 3D model of the 'lemon' type of curvature singularity, in the form of the lines of curvature of an ellipsoid:

I would like to have a similar understanding of the behaviour of the lines of curvature around the star and monstar umbilical points, through a specific 2D surface embedded in real 3D space. (This is with a view to making 3D printable models of the three so I can examine them in person to help with the visualization.)

What are clean ways to get such models?

E.P.
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  • I can't say this with complete certainty, but I'll bet that the "star" pattern arises in the case of the monkey-saddle. – John Hughes Nov 09 '17 at 11:27
  • @JohnHughes Yeah, I can imagine. I'm primarily interested in understanding the difference between the lemon and the monstar, but I'd like the set. – E.P. Nov 09 '17 at 11:28

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