Monge surfaces and planar geodesic foliations

David Brander*, Jens Gravesen

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

141 Downloads (Pure)

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
Original languageEnglish
JournalJournal of Geometry
Volume109
Issue number4
Number of pages14
ISSN0047-2468
DOIs
Publication statusPublished - 2018

Keywords

  • Monge surface
  • Planar geodesic
  • Developable surface

Cite this

@article{410cbf028293479a842fcfe21f392b97,
title = "Monge surfaces and planar geodesic foliations",
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",
author = "David Brander and Jens Gravesen",
year = "2018",
doi = "10.1007/s00022-018-0413-7",
language = "English",
volume = "109",
journal = "Journal of Geometry",
issn = "0047-2468",
publisher = "Springer Basel AG",
number = "4",

}

Monge surfaces and planar geodesic foliations. / Brander, David; Gravesen, Jens.

In: Journal of Geometry, Vol. 109, No. 4, 2018.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Monge surfaces and planar geodesic foliations

AU - Brander, David

AU - Gravesen, Jens

PY - 2018

Y1 - 2018

N2 - 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

AB - 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

KW - Monge surface

KW - Planar geodesic

KW - Developable surface

U2 - 10.1007/s00022-018-0413-7

DO - 10.1007/s00022-018-0413-7

M3 - Journal article

VL - 109

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 4

ER -