Supercyclidic Nets
Alexander I. Bobenko, Emanuel Huhnen-Venedey, Thilo Rörig · International Mathematics Research Notices · 2016
Supercyclides are surfaces with a characteristic conjugate parameterization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the resulting patchworks “supercyclidic nets” and show that every Q-net in |$\mathbb {R}{\rm P}^3$| can be extended to a supercyclidic net. The construction is governed by a multidimensionally consistent three-dimensional system. One essential aspect of the theory is the extension of a given Q-net in |$\mathbb {R}{\rm P}^N$| to a system of circumscribed discrete torsal line systems. We present a description of the latter in terms of projective reflections that generalizes the systems of orthogonal reflections which govern the extension of circular nets to cyclidic nets by means of Dupin cyclide patches.