Multi-dimensional consistency of principal binets

TL;DR

This paper introduces principal binets as a new class of discretized curvature surfaces on square lattices, demonstrating their multi-dimensional consistency and relation to discrete orthogonal systems.

Contribution

It establishes principal binets as a novel integrable discretization extending to higher dimensions and connects them to discrete orthogonal coordinate systems.

Findings

Principal binets are a new discretization of curvature line surfaces.

They form a multi-dimensional consistent integrable system.

They relate to discrete orthogonal coordinate systems and confocal quadrics.

Abstract

Principal binets are a discretization of curvature line parametrized surfaces defined on the vertices and faces of the square lattice $\Z^2$. They generalize the previously established discretizations given by circular nets, conical nets, and principal contact element nets. We show that principal binets constitute a discrete integrable system in the sense of multi-dimensional consistency. In particular, they generalize to higher-dimensional square lattices $\Z^N$. We also discuss relations to the notion of discrete orthogonal coordinate systems as previously established for discrete confocal quadrics.