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?
On
Two remarks:
a) I calculated some time ago exactly what you ask for and found e.g. a condition that the osculating conic at a given point is an ellipse (resp. parabola, hyperbola), this is the case 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. See https://mathoverflow.net/questions/28044/osculating-conics-and-cubics-and-beyond
b) In the meantime Wolfram Alpha has implemented a function OsculatingConic: https://resources.wolframcloud.com/FunctionRepository/resources/OsculatingConic/
On
Conic can osculate a curve at least up to $4$-th order of contact (multiplicity of $5$).
\begin{align} 0 &= 9(y'')^4X^2+6(y'')^2 y''' X(Y-y' X) \\ & \quad +[3y'' y^{(4)}-4(y''')^2](Y-y'X)^2-18(y'')^3 (Y-y'X) \end{align}
$$ 0=\det \begin{pmatrix} (X-x)^2 & 0 & x'^2 & 3x'x'' & 3x''^2+4x'x''' \\ 2(X-x)(Y-y) & 0 & 2x'y' & 3x'y''+3x''y' & 6x''y''+4x'y'''+4x'''y' \\ (Y-y)^2 & 0 & y'^2 & 3y'y'' & 3y''^2+4y'y''' \\ 2(X-x) & x' & x'' & x''' & x'''' \\ 2(Y-y) & y' & y'' & y''' & y'''' \end{pmatrix}$$
The generic equation of a conic can be written as: $$ ax^2+bxy+cy^2+dx+ey+f=0. $$
Suppose your curve is given as $(x(t),y(t))$ and you want to find its osculating conic at $t=t_0$. If you define $$ F(t)=ax(t)^2+bx(t)y(t)+cy(t)^2+dx(t)+ey(t)+f $$ then the following conditions must hold: $$ F(t_0)=0,\quad F'(t_0)=0,\quad F''(t_0)=0,\quad F'''(t_0)=0,\quad F''''(t_0)=0. $$ Solve the above system of equations to find coefficients $a,\dots ,f$.