The B\"acklund Transform of Principal Contact Element Nets

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.