Focal Surfaces of Characteristic-Line Congruences

Classify the focal surfaces and induced surface transformations associated with congruences of characteristic lines arising from algebraic tables in projective contact three-space.

Background

For a general algebraic table, its characteristic lines form a congruence whose envelope may have multiple focal branches and whose lines may touch the focal surface at several points. Beyond the table and its polar surface, the paper leaves the possible focal geometries and resulting conjugate-net transformations open.

References

Apart from the table and its polar, what sorts of focal surfaces are associated with a congruence of characteristic lines along an algebraic table in $(\mathbb{RP}3,\xi)$? What sorts of surface transformations and sequences of conjugate nets (see \S 5 of ) can arise from such congruences composed of characteristic lines?

— Outer Contact Billiards  (2608.19393 - Caliz et al., 19 Aug 2026) in Section 6, “Open questions,” item 6