Well known background: Consider a differentiable surface in $\mathbb{R}^3$ and a point $p$; a normal plane at $p$ is a plane containing the normal vector. It is a classical result that the curvature of the intersection of a normal plane in $p$ and the surface (the normal section) has a curvature in $p$ depending on the normal plane and that maximum and minimum value of this curvature (known as principal curvatures $k_1$ and $k_2$) occur at perpendicular directions of the normal plane.
The dependence of the curvature in the normal section on the angle of the normal plane is given by $k(\alpha) = k_1 \cos^2 \alpha + k_2 \sin^2 \alpha$ with $\alpha$ being the angle with respect to the direction where the principal curvature $k_1$ occurs. Sometimes this equation is called Euler’s equation (one of many).
Now we consider in the normal section the osculating conic at $p$ (in the sense of question Osculating conics and cubics and beyond). Thus we don’t have any more an osculating circle defining the curvature $k(\alpha)$ in each normal section, but an osculating conic with in general two focal points, semi-axis and eccentricity.
Some questions seem to be very natural, but I couldn’t find any results:
What is the surface built by the set of the osculating conics at a point $p$ (if we vary the normal section)? It does not seem to be an osculating quadric at point $p$ for the reason that the osculating conic in the normal section does not need to be symmetric with respect to the normal vector in $p$, whereas normal sections of an osculating quadric are (or am I mistaken here?).
How do the characteristic parameters (take e.g. the eccentricity) of the osculating conics depend on the normal plane, is there an analogous equation to the one of Euler for the curvature (of the osculating circle)?
Is there an explicit description of this surface in terms of the partial derivations of $f$, if the surface is given as a graph $f(x,y)$ (let’s assume wlog $f_{x}= f_{y} =0$).
If all the osculating conics in the normal sections are parabolic (in the sense of the above mentioned question), what does this mean for the original surface at point $p$?
