Discrete curve theory in space forms: planar elastic and area-constrained elastic curves

TL;DR

This paper develops a theory of discrete elastic and area-constrained elastic curves in 2D space forms, extending smooth geometric concepts to discrete settings and analyzing their properties and flows.

Contribution

It introduces a novel discrete curvature framework in space forms, including flow dynamics and hierarchy characterization of elastic curves.

Findings

Discrete elastic curves are characterized within a new hierarchy.

Discrete flows from Bäcklund transformations are analyzed.

The theory extends classical smooth curve properties to discrete space forms.

Abstract

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from B\"acklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.