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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.

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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.

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  • $\begingroup$ I can see how that makes sense. Any idea how to go about determining the osculating conic given a parameterization of the plane curve by $\textbf{c}(t)$? $\endgroup$
    – R. Burton
    Commented Feb 26, 2019 at 22:39
  • $\begingroup$ osculating from the Latin for "kissing". Not to be confused with "oscillating". $\endgroup$
    – DragonLord
    Commented Feb 15, 2020 at 21:17

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