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

Tim Hoffmann, Jannik Steinmeier, Gudrun Szewieczek

TL;DR

The paper develops a theory of discrete elastic and area-constrained elastic curves in 2D space forms, extending the discrete curvature framework from Euclidean to ambient spaces of constant curvature. It builds a comprehensive framework using light-cone and matrix models, a discrete Frenet-type formalism, and an associated family to generate and relate discrete curves. Central results identify elastic curves as 2-invariant and area-constrained elastic curves as 3-invariant under sequences of Bäcklund transformations, thereby establishing a hierarchical, integrable-flow picture anchored in Darboux/Bäcklund theory. The work connects to broader discrete differential geometry themes, including skew parallelogram nets and discrete isothermic structures, and lays groundwork for further exploration of discrete elastic curves in non-Euclidean ambient geometries.

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 Bäcklund 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.

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

TL;DR

The paper develops a theory of discrete elastic and area-constrained elastic curves in 2D space forms, extending the discrete curvature framework from Euclidean to ambient spaces of constant curvature. It builds a comprehensive framework using light-cone and matrix models, a discrete Frenet-type formalism, and an associated family to generate and relate discrete curves. Central results identify elastic curves as 2-invariant and area-constrained elastic curves as 3-invariant under sequences of Bäcklund transformations, thereby establishing a hierarchical, integrable-flow picture anchored in Darboux/Bäcklund theory. The work connects to broader discrete differential geometry themes, including skew parallelogram nets and discrete isothermic structures, and lays groundwork for further exploration of discrete elastic curves in non-Euclidean ambient geometries.

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 Bäcklund 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.
Paper Structure (18 sections, 21 theorems, 122 equations, 14 figures)

This paper contains 18 sections, 21 theorems, 122 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Discrete Curve Theory in space forms
  3. Various models for space forms
  4. Light cone model
  5. Matrix model
  6. Conversion of the models
  7. Basic differential geometric notions
  8. Light-cone model
  9. Matrix model
  10. Constructions for discrete curves in space forms
  11. Frenet-type equations for discrete curves in 2-dimensional space forms
  12. Associated family of curves
  13. Discrete constrained elastic curves in space forms
  14. Flows and invariant curves
  15. Bäcklund transformations and $n$-invariant curves
  16. ...and 3 more sections

Key Result

Proposition 7

In Euclidean space we have $H_0=\bm 1+\frac{\eta}{2}\kappa_0 \bm k$. In non-Euclidean space we have $H_0=\bm 1+ \frac{\zeta}{2}\kappa_0 F_0$.

Figures (14)

  • Figure 1: Notation convention used for discrete curves.
  • Figure 2: Arc-length parametrized discrete curves in Euclidean, hyperbolic and elliptic space with their geodesic edges (orange), curvature circles (dark blue) and double-curvature circles (light blue).
  • Figure 3: Curves obtained from $\kappa(t_i)=a i$ for some constant $a\in\mathbb{R}$ form discrete Euler spirals/clothoids in Euclidean space $\mathbb{E}^2$, hyperbolic space $\mathbb{H}^2$ (the Poincaré disc) and spherical space $\mathbb{S}^2$.
  • Figure 4: A discrete elastic curve in Euclidean space and a discrete area-constrained elastic curve in hyperbolic space with their directrices (dotted) and some double-curvature circles (gray). The latter intersect the directrix at a constant angle.
  • Figure 5: A discrete (area-constrained) elastic curve in Euclidean space: its curvature and orthogonal (squared tangential) distance to the corresponding directrix are proportional (see Corollary \ref{['cor:ConstrElasticProp']}).
  • ...and 9 more figures

Theorems & Definitions (65)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7
  • proof
  • Definition 8
  • Proposition 9
  • ...and 55 more