When Is a Periodic Function the Curvature of a Closed Plane Curve?

The Stability and Evolution of Curved Domains Arising from One-Dimensional Localized Patterns

“…In particular (cf. [1]), if κ(s) is periodic with minimal period ρ κ , then the curve Γ κ whose curvature is κ closes up in [0, nρ κ ], n > 1, if and only if there exists an integer m such that From the phase plane of (3.29), shown in Figure 3, the symmetry of large-amplitude orbits implies that ρκ 0 κ(s) ds = 0, and consequently these orbits will never generate closed planar curves. The small-amplitude orbits seen inside the homoclinic of Figure 3(a) do have nonzero values of ρκ 0 κ(s) ds.…”

Patched peakon weak solutions of the modified Camassa–Holm equation

“…The angle function is then defined as We now present a criterion on θ that guarantees that the curve is closed. With a modification of results in [2], we have Then, the total length of γ must be n 1 T so that n 1 |n for some integer n 1 . By the definition of γ(s), θ is the incline angle of the tangent vector and hence n 1 θ(T ) is a multiple of 2π since the curve is closed, or n|mn 1 .…”

“…In the case θ(T ) ∈ 2πZ, one can refer to [2] for examples where the curves are not closed. Now, we show that some patched periodic traveling peakon weak solutions of (5.8) may correspond to loops with cusps.…”

Characterisation of cylindrical curves

“…In [ 1 ] a necessary and sufficient condition was given for a planar curve of periodic curvature to be closed. Since a cylinder is a developable surface this result immediately gives a sufficient condition for a cylindrical curve to be closed provided we replace (signed) curvature by geodesic curvature:…”