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).