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?