With this question I would like to kindly ask for feedback or general pointers to even remotely related works in regards to a challenge I face.
Given a smooth surface $S$ $:\mathbb{R}^2\rightarrow \mathbb{R}^3$ and a plane $P$ defined using its attachment point $r$ and normal $n$ their intersection defines the planar curve $C:[0,1]\rightarrow \mathbb{R}^3$. If we displace the plane by a "small" $d$ vector to get $P^*$ a new cross section curve $C^*$ is formed.
Considering $C$ and $C^*$ as curves embedded in the planes $P$ and $P^*$ the effect of the displacement by $d$ can be associated to a mapping $\phi: \mathbb{R}^2\rightarrow\mathbb{R}^2$ that transforms $C$ to $C^*$.
My goal is to be able to calculate derivatives of the $\phi$ mapping with respect to the plane parameters (in this case the displacement $d$) given a local characterization of the geometry of the surface along the plane.
The idea I was considering was to rely on Christoffel-symbols of the second kind computed for the known surface $S$ and somehow express $\phi$ by means of geodesics starting from the points of $C$ along directions that correspond to projections of $d$ on the local tangent planes. As Christoffel-symbols should capture the local behavior of the surface I would hope that shape changes due to the out of plane motion could be explained this way for small displacements.
I have to admit that I only possess a very basic understanding of differential geometry, but I believe that the problem I try to solve can be addressed in this framework, and would welcome any remark or suggestion. The principal motivation behind the problem is the alignment of planar curves to a known 3D surface, where the curves are observed by a planar cross section imaging process.
Thanks a lot!
The surface $S$ (cone) and its planar cross sections
The problem is that knowing $C$ and $C'$, you have to associate to a point of $C$ some point of $C'$. One way to do that is to take, for a point $p \in C$, the "nearest" intersection of the normal line at $p$ with $C'$. That'll work OK for really smooth surfaces and really well-chosen slices. But in general, you'll have problems with "focal points" of $C$, and with the fact that the slices of a smooth surface need not be smooth curves in the plane. In short: any solution your come up with is likely to have frequent enough problems that it'll be really tough to use in practice. (I say this as someone who's studied and taught both geometry and topology, got a Ph.D. in math, and have spent the last 30 years doing computer graphics, and have tried to do things like this repeatedly and without success. You can get good results...but you can also always find examples that break whatever you're trying.)