Discrete Laplace Cycles of Period Four

TL;DR

This paper investigates discrete conjugate nets with a Laplace sequence of period four, revealing geometric properties and providing explicit construction methods for these special cyclic nets.

Contribution

It introduces the concept of discrete Laplace cycles of period four, analyzes their properties, and offers two explicit construction methods for these nets.

Findings

Opposite nets in the cycle share equal osculating planes in different directions.

Connecting lines of corresponding points form a discrete W-congruence.

Properties of discrete Laplace cycles of period four are characterized and constructed explicitly.

Abstract

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.