Supercyclidic Nets

Abstract

Supercyclides are surfaces with a characteristic conjugate parametrization 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 RP 3 can be extended to a supercyclidic net. The construction is governed by a multidimensionally consistent 3D system. One essential aspect of the theory is the extension of a given Q-net in RP 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.2010 Mathematics Subject Classification. 51A05, 53A20, 37K25, 65D17.

Discrete line complexes and integrable evolution of minors

“…In fact, this construction reveals that any 'elementary cube' of a fundamental line complex should be regarded as being embedded in a classical spatial point-line configuration (15 4 20 3 ) of 15 points and 20 lines [20]. It is observed that some of the theorems about fundamental line complexes set down in this section turn out to be important in the construction of supercyclidic nets [21].…”