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We consider a series of optimal control problems with 2-dimensional controllying in an arbitrary convex compact set $\Omega$. The considered problems arewell studied for the case when $\Omega$ is a unit disc, but barely studied forarbitrary $\Omega$. We derive extremals to these problems in general case byusing machinery of convex trigonometry, which allows us to do this identicallyand independently on the shape of $\Omega$. The paper describes geodesics in(i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii)left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2),SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane withdistance function given by $\Omega$; (iv) a series of "yacht problems"generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Sheppproblem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamicmotion problem.

Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry