  A new convenient method of describing flat convex compact sets is proposed.
It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently,
this method may be very useful for explicit description of solutions of optimal
control problems with two-dimensional control. Using this method a series of
sub-Finsler problems with two-dimensional control lying in an arbitrary convex
set $\Omega$ is investigated. Namely, problems on the Heisenberg, Engel, and
Cartan groups and also Grushin's and Martinet's cases are considered. A
particular attention is paid to the case when $\Omega$ is a polygon.

Optimization and Control;Differential Geometry