The B\"acklund Transform of Principal Contact Element Nets

TL;DR

This paper explores the geometric properties of the Bäcklund transform applied to principal contact element nets, establishing conditions for its existence, providing a construction method, and confirming the validity of Bianchi's Permutation Theorem in a discrete context.

Contribution

It introduces a new elementary construction of the Bäcklund transform for principal contact element nets and proves its correctness, extending classical results to a discrete geometric setting.

Findings

Bäcklund transform exists iff the net has constant negative Gaussian curvature

Elementary construction of the Bäcklund transform is validated

Bianchi's Permutation Theorem holds in the discrete setting

Abstract

We investigate geometric aspects of the the B\"acklund transform of principal contact element nets. A B\"acklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the B\"acklund transform and prove its correctness. Finally, we show that Bianchi's Permutation Theorem remains valid in our discrete setting.