Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with dupin cyclides. Geometriae Dedicata 159

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimen-sional cyclidic net is a piecewise smooth C1-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 curva-ture line parametrized surfaces as circular, conical and principal con-tact 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.