Principal binets

TL;DR

This paper introduces principal binets, a new discretization of principal curvature line parametrizations using polar pairs of line congruences, expanding the framework of discrete differential geometry with invariance and consistency properties.

Contribution

It proposes principal binets, a novel discretization method for principal curvature lines, generalizing existing circular and conical nets with invariance and higher-dimensional consistency.

Findings

Principal binets generalize discrete conjugate nets with orthogonal edges.

All discretizations satisfy Lie, M"obius, or Laguerre invariance.

The methods are consistent in higher-dimensional lattice extensions.

Abstract

Conjugate line parametrizations of surfaces were first discretized almost a century ago as quad meshes with planar faces. With the recent development of discrete differential geometry, two discretizations of principal curvature line parametrizations were discovered: circular nets and conical nets, both of which are special cases of discrete conjugate nets. Subsequently, circular and conical nets were given a unified description as isotropic line congruences in the Lie quadric. We propose a generalization by considering polar pairs of line congruences in the ambient space of the Lie quadric. These correspond to pairs of discrete conjugate nets with orthogonal edges, which we call principal binets, a new and more general discretization of principal curvature line parametrizations. We also introduce two new discretizations of orthogonal and Gauss-orthogonal parametrizations. All our discretizations are subject to the transformation group principle, which means that they satisfy the corresponding Lie, M\"obius, or Laguerre invariance respectively, in analogy to the smooth theory. Finally, we show that they satisfy the consistency principle, which means that our definitions generalize to higher dimensional square lattices. Our work expands on recent work by Dellinger on checkerboard patterns.