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Geometriae Dedicata

, Volume 159, Issue 1, pp 207–237 | Cite as

Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides

  • Alexander I. Bobenko
  • Emanuel Huhnen-VenedeyEmail author
Original Paper

Abstract

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth C 1-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit description of cyclidic nets is given and their relation to the established discretizations of curvature line parametrized surfaces as circular, conical and principal contact element nets is explained. We introduce 3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate systems and investigate them in detail. Our considerations are based on the Lie geometric description of Dupin cyclides. Explicit formulas are derived and implemented in a computer program.

Keywords

Discrete differential geometry Curvature line parametrized surfaces Orthogonal coordinate systems Dupin cyclides Lie geometry Cyclidic nets Circular nets Principal contact element nets Conical nets 

Mathematics Subject Classification (2000)

51B10 51B25 53A30 37K25 52C26 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 8-3Technische Universität BerlinBerlinGermany

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