  We construct Peano curves $\gamma : [0,\infty) \to \mathbb{R}^2$ whose
"footprints" $\gamma([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent
to a common continuous line field on the punctured plane $\mathbb{R}^2
\setminus \{\gamma(0)\}$. Moreover, these boundaries can be taken
$C^\infty$-close to any prescribed smooth family of nested smooth Jordan curves
contracting to a point.
