Elastic Curves via Geometric Mechanics
2026-07-31 • Graphics
Graphics
AI summaryⓘ
The authors study elastic curves, which are the shapes thin rods take when balanced, using a new way based on geometry and mechanics. They focus on an old but less-known idea that these curves minimize length while keeping area and volume fixed. They apply this idea to polygonal curves, creating a discrete version that keeps key properties without complicated bending calculations. This approach also connects to advanced math structures that help understand how these curves change dynamically.
elastic curvesbending energyisoperimetric problempolygonal curvesrigid body motionsmomentum mapMarsden–Weinstein reductionpre-symplectic structureHamiltonian dynamicsvortex filaments
Authors
Oliver Gross, Rohit Jammula, Albert Chern
Abstract
Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden--Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg--de Vries flows.