  Given a smooth curve $\gamma$ in some $m$-dimensional surface $M$ in
$\mathbb{R}^{m+1}$, we study existence and uniqueness of a flat surface $H$
having the same field of normal vectors as $M$ along $\gamma$, which we call a
flat approximation of $M$ along $\gamma$. In particular, the well-known
characterisation of flat surfaces as torses (ruled surfaces with tangent plane
stable along the rulings) allows us to give an explicit parametric construction
of such approximation.
