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The osculating circle at a point of a smooth plane curve can be obtained by considering three points on the curve and the circle defined by them. When the three points approach $P$, the circle becomes the osculating circle at $P$.

Generalizing this method to a conic defined by five points on a curve, one obtains an 'osculating conic' at each point of a smooth plane curve. It is pretty straightforward to show that this osculating conic is an ellipse (resp. parabola, hyperbola) if $y'' y^{(4)} > \frac{5}{3}y'''^2$ (resp. =, <).

Thus one can classify points on a plane curve, e.g. all points on $y=e^x$ are hyperbolic (which is not obvious to the 'naked eye'). The aforementioned criterion is an affine differential invariant.

I'm wondering why these 'osculating conics' seem to be relatively unknown (they would give a nice textbook example connecting elementary calculus and linear algebra - you can find the criterion above in one handwritten page, starting with the Taylor expansion) and whether there are any applications (generalization of evolute; can you get the curve back from the focal curve of the osculating conics)?

Furthermore it would be interesting to know whether there are any results for 'osculating cubics' (nine points define a cubic plane curve) or 'osculating quartics' (fourteen points define a quartic plane curve) or for 'osculating quadrics' (would in general yield another classification of points on a surface beyond the usual elliptic/parabolic/hyperbolic one).

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    $\begingroup$ Your description of local higher order affine geometric invariants of a curve is very nice, but, for at least some of us, a geometric property that is not "visible to the naked eye" is just not really geometric. Anything beyond second order is too subtle. Also, I believe that the lack of good guiding examples or applications has made local affine differential geometry a difficult subject to sell. $\endgroup$
    – Deane Yang
    Commented Jun 13, 2010 at 18:44
  • $\begingroup$ One tautological fact is that for each $d > 0$, osculation induces a canonical map from your curve to a the Hilbert scheme of degree $d$ plane curves. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 14, 2010 at 2:23
  • $\begingroup$ Just to comment: IIRC the concept of "aberrancy" (a gemoetric way to interpret the third derivative, much like curvature for the second derivative), like in these papers: jstor.org/stable/2690245 & jstor.org/stable/2320414 , ties very intimately to the concept of the (pen)osculating conic; maybe you should look into these? $\endgroup$
    – J. M. isn't a mathematician
    Commented Aug 12, 2010 at 7:57

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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

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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

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