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).