Monatshefte für Mathematik

, Volume 180, Issue 4, pp 693–712 | Cite as

Peano curves with smooth footprints

  • Jairo Bochi
  • Pedro H. Milet


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


Space-filling curves Smoothness Convexity Line fields 

Mathematics Subject Classification

26A30 26E10 


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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

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