Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides

Details Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides

2011-01-31

arxiv.org/abs/1101.5955v1

Mathematics

Differential Geometry

39 pages, 31 figures

Scientific paper

10.1007/s10711-011-9653-5

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.

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