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Journal of Geometry

, 109:4 | Cite as

Monge surfaces and planar geodesic foliations

  • David Brander
  • Jens Gravesen
Article

Abstract

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally defined objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix.

Keywords

Monge surface planar geodesic developable surface 

Mathematics Subject Classification

Primary 53A05 Secondary 53C12 53C22 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark

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