I am looking for information and relevant terminology for a phenomenon related to the geometry of sublevel sets of functions. I believe that the possibly related fields are Morse theory and algebraic geometry, although there are probably more elementary ways to describe this phenomenon.
The motivation for my question is the following picture:
This is numerics from some problem I was working on. It is basically a plot of the sublevel sets of some function $f(x,y)$. This is a semi-algebraic set, with a special point in the middle, which locally looks like a union of two cones. I am interested in understanding more about this special point.
I have already understood that this point should probably correspond to a critical point of $f$ where the Hessian of $f$ should also be degenerate. The reason for this is that if the Hessian is nondegenerate, then $f$ can have at most one stable and one unstable direction, and here it seems like it has two stable directions and two unstable directions (corresponding to the two cones).
But I am interested in finding some relevant terminology that would help me look up some basic properties of such points (the only Morse theory I learned was for nondegenerate critical points) - perhaps there are some terms from the world of semi-algebraic sets or Morse theory that describe these "higher order cones" and some of their properties?
My main interest is being able to characterize these cones more easily (how many stable/unstable directions, how many cones are "meeting", etc.). This should probably depend on the higher order differentials of $f$, but I am not sure how exactly. I can sometimes have cones of even "higher" order (three cones or more meeting):
and I would like to understand those as well. I am also interested in higher dimensional cones, when the function $f$ takes more than two variables.
Anyway, any help with finding relevant references would be very helpful. Thanks in advance!