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

摘要

Cyclidic nets are introduced as discrete analogs of curvature lineparametrized surfaces and orthogonal coordinate systems. A 2-dimensionalcyclidic net is a piecewise smooth $C^1$-surface built from surface patches ofDupin cyclides, each patch being bounded by curvature lines of the supportingcyclide. An explicit description of cyclidic nets is given and their relationto the established discretizations of curvature line parametrized surfaces ascircular, conical and principal contact element nets is explained. We introduce3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinatesystems and investigate them in detail. Our considerations are based on the Liegeometric description of Dupin cyclides. Explicit formulas are derived andimplemented in a computer program.