Cartan ribbonization and a topological inspection

Matteo Raffaelli*, Jakob Bohr, Steen Markvorsen

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.
Original languageEnglish
Article number20170389
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume474
Issue number2220
Number of pages17
ISSN1364-5021
DOIs
Publication statusPublished - 2018

Keywords

  • Developable surfaces
  • Ribbons
  • Rolling
  • Topological inspection

Cite this

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title = "Cartan ribbonization and a topological inspection",
abstract = "We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.",
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Cartan ribbonization and a topological inspection. / Raffaelli, Matteo; Bohr, Jakob; Markvorsen, Steen.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, No. 2220, 20170389, 2018.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Cartan ribbonization and a topological inspection

AU - Raffaelli, Matteo

AU - Bohr, Jakob

AU - Markvorsen, Steen

PY - 2018

Y1 - 2018

N2 - We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.

AB - We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.

KW - Developable surfaces

KW - Ribbons

KW - Rolling

KW - Topological inspection

U2 - 10.1098/rspa.2017.0389

DO - 10.1098/rspa.2017.0389

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JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

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