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

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.

39 pages, 30 figures; Theorem 2.7 has been reformulated, as a normalization factor in formula (2.4) was missing. The corresponding formulations have been adjusted and a few typos have been corrected

References in corpus (2)

• On organizing principles of Discrete Differential Geometry. Geometry of spheres
• Uniform convergence of discrete curvatures from nets of curvature lines

Cited by in corpus (8)

• Discretization of asymptotic line parametrizations using hyperboloid patches
• Discrete channel surfaces
• Ribaucour coordinates
• On Weingarten transformations of hyperbolic nets
• Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics
• Classification of Dupin Cyclidic Cubes by Their Singularities
• Shapes of surfaces that contain a great and a small circle through each point
• Self-intersections of surfaces that contain two circles through each point