Discrete Koenigs nets, inscribed quadrics and autoconjugate curves

TL;DR

This paper extends the understanding of discrete Koenigs nets by introducing higher-dimensional inscribed quadrics, analyzing Koenigs d-grids, and establishing a connection with pairs of autoconjugate curves, advancing discrete differential geometry.

Contribution

It generalizes touching inscribed conics to higher-dimensional quadrics and links Koenigs d-grids with autoconjugate curves, revealing new geometric properties.

Findings

Existence of higher-dimensional inscribed quadrics for Koenigs nets.

Koenigs d-grids have a unique inscribed quadric tangent to all parameter spaces.

Bijection established between Koenigs d-grids and pairs of autoconjugate curves.

Abstract

Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.