Supercyclidic nets

TL;DR

This paper introduces supercyclidic nets, a novel class of discrete surface patches based on supercyclides, extending Q-nets in projective space through a multidimensional consistent system governed by projective reflections.

Contribution

It develops a new framework for extending Q-nets to supercyclidic nets using a multidimensional system and projective reflections, generalizing previous theories of circular and cyclidic nets.

Findings

Every Q-net in $ ext{RP}^3$ can be extended to a supercyclidic net.

The extension is governed by a multidimensional consistent 3D system.

The theory generalizes orthogonal reflection systems for circular 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 $\mathbb{R}P^3$ 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 $\mathbb{R}P^N$ 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.