Is there a form of curvature for a plane curve which is invariant under uniform scaling?
Ideally, I am looking for a way to characterize the effective 'local eccentricity' of a plane curve so that [geometrically] similar curves have identical curvature at corresponding points - i.e. every circle has the same curvature regardless of radius, every parabola, every hyperbola with the same eccentricity, etc.
What you are after is probably the eccentricity of the osculating conic.
Note that a circle is defined by three points; this is why the osculating circle involves derivatives up to the second order.
Similarly, a general conic requires five points, and the osculating conic will take fourth derivatives.