Osculating conics and cubics and beyond

These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important references will be:

Topological invariants of plane curves and caustics. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5. American Mathematical Society, Providence, RI, 1994.

More precisely, what was studied are the points of the curve, where the level of its tangency with (say) conics is higher than expected. I guess these are exactly the points that (using your terminology) separate the elliptic part of the curve from the hyperbolic part.

The key words for this research are Extactic points (terminology proposed by D. Eisenbud). Using google scholar you can find a complete text of Arnol'd, called:

Remarks on the extatic points of plane curves, V.I. Arnold - The Gelfand Mathematical Seminars, 1993-1995.

This article contains some generalisations of the four vertex theorem. http://en.wikipedia.org/wiki/Four-vertex_theorem

One more nice reference is a paper of Tabachnikov and Timorin. https://arxiv.org/abs/math/0602317

My REU project was about this topic.

We found some interesting things about when osculating cubics are unique (plot twist: not always), as well as a formula for the osculating conic of a smooth plane curve (at a non-flex point). Here is the formula (copied from a software package I wrote, sorry for the mess),

Given F(x,y):=(higher order terms) +a*x^4+b*x^3*y+c*x^2*y^2+d*x*y^3+e*y^4+f*x^3+g*x^2*y+h*x*y^2+i*y^3+j*x^2+k*x*y+l*y^2+m*x+n*y

The osculating conic of V(F) at (0,0) is given by

OscConic(a,b,c,d,e,f,g,h,i,j,k,l,m,n)=m*x+n*y+(j*x^2+k*x*y+l*y^2)+(-((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)^2)/(j*n^2-k*m*n+l*m^2)^3*(m*x+n*y)^2+(a*n^4-b*m*n^3+c*m^2*n^2+(-d)*m^3*n+e*m^4)/(j*n^2-k*m*n+l*m^2)^2*(m*x+n*y)^2+(((x*(k*m-2*j*n)+y*(2*l*m-k*n))*((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)-(x*(3*f*n^2-2*g*m*n+h*m^2)+y*(g*n^2-2*h*m*n+3*i*m^2))*(j*n^2-k*m*n+l*m^2))*(m*x+n*y))/(j*n^2-k*m*n+l*m^2)^2

The formula comes from the paper we wrote (look at Lemma 2.22 and 2.24) here

Points of Ninth Order on Cubic Curves