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For each right-angled hexagon in the hyperbolic plane, we construct aone-parameter family of right-angled hexagons with a Lipschitz map between anytwo elements in this family, realizing the smallest Lipschitz constant in thehomotopy class of this map relative to the boundary. As a consequence of thisconstruction, we exhibit new geodesics for the arc metric on theTeichm{\"u}ller space of an arbitrary surface of negative Euler characteristicwith nonempty boundary. We also obtain new geodesics for Thurston's metric onTeichm{\"u}ller spaces of hyperbolic surfaces without boundary.

Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichm{\"u}ller spaces