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The osculating circle at a point of a smooth plain 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 plain curve. It is pretty straight forward 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 plain curve, e.g. all points on $y=e^x$ are hyperbolic (which is not obvious to the 'naked eye'). The mentionned criterion is an affine differential invariant.

I'm wondering why these 'osculating conics' seem to be relativley unknown (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 foci 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 plain curve) or 'osculating quartics' (fourteen points define a quartic plain curve) or for 'osculating quadrics' (would in general yield another classification of points on a surface than the usual elliptic/parabolic/hyperbolic one).

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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. –  Deane Yang Jun 13 '10 at 18:44
    
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. –  S. Carnahan Jun 14 '10 at 2:23
    
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? –  J. M. Aug 12 '10 at 7:57
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up vote 6 down vote accepted

These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important refferences 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 studdied are the points of the curve, where the level of it tangency with (say) conics is higher than expected. I guess these are exactly the points that (using your terminology) separate elliptic part of the curve form hyperbolic.

The key words for these research are Extactic points (therminology proposed by D. Esenbud). 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.

These article contains some genearlisations of four vertex theorem. http://en.wikipedia.org/wiki/Four-vertex_theorem

One more nice refference is a paper of Tabachnikov and Timorin http://arxiv.org/PS_cache/math/pdf/0602/0602317v2.pdf

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