Smooth and Discrete Cone-Nets

Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of that transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in the classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. We direct special emphasis towards smooth and discrete tractrix surfaces which are characterized as principal cone-nets with constant geodesic curvature along one family of parameter curves.

Automated summary

Cone-nets are conjugate nets on a surface where each individual curve of a parameter curve family is in tangential contact with a cone. The corresponding conjugate curve network is projectively invariant and can be characterized by specific transformations. This study focuses on the properties of the transformation theory and demonstrates how different surface classes can be represented within this framework. Cone-nets are examined in both the smooth setting of differential geometry and in a consistent discretization with counterparts to relevant statements and notions of the smooth setting. Special attention is given to smooth and discrete tractrix surfaces, which are identified as principal cone-nets with constant geodesic curvature along one family of parameter curves.