  In this paper, we show that the $C^1$-differentiability of the norm of a
two-dimensional normed space depends only on distances between points of the
unit sphere in two different ways.
  As a consequence, we see that any isometry between the spheres of normed
planes $\tau:S_X\to S_Y$ is linear, provided that there exist linearly
independent $x,\overline{x}\in S_X$ where $S_X$ is not differentiable and that
$S_X$ is piecewise differentiable.
  We end this work by showing that the isometry $\tau:C_X\to C_Y$ is linear
even if it is not an isometry between spheres: every isometry between (planar)
Jordan piecewise $C^1$-differentiable convex curves extends to $X$ whenever $X$
and $Y$ are strictly convex and the amount of non-differentiability points of
$S_X$ and $S_Y$ is finite and greater than 2.
