I wrote my Bachelor's thesis at TU Berlin under supervision of Dr. Jan Techter and Prof. Dr. Alexander Bobenko.

Circular sections of quadrics

Generically, a quadric $Q$ posses two one-parameter families of parallel planes $\Sigma_{\pm}(\lambda)$ intersecting it in circular sections, where $\lambda$ denotes a real parameter. These circles constitute a net on $Q$, i.e. they form a pair of one-parameter families of curves $\beta_{\pm}(\lambda)$, such that for every point of $Q$ there exists exactly one curve from each of the two families through that point.

In this thesis, we derive the explicit equations of these parallel planes using the absolute quadric of Euclidean geometry in a projective setting. Then we prove that this net of circles on a quadric is diagonally related to the net of curvature lines of the quadric, a condition referenced in this paper. Finally, we present an explicit family of isometric deformations (for one-sheeted and two-sheeted hyperboloids) that preserves the circles, the curvature lines and their diagonal relation.

For a nice reference to the work in this thesis (especially for the case of ellipsoids), check out this talk by Prof. Wolfgang Schief.