Peano curves with smooth footprints
  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. 
Jairo Bochi, Pedro H. Milet
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