A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the eccentricity of the osculating conic at that point.
While I have been able to find many, many methods for finding osculating circles, I cannot seem to find any for osculating conics. Is there a particular formula for finding the osculating conic given the parameterization of a plane curve?