Integrable Osculating Plane Distributions

The Osculating Plane of a Space Curve - Synthetic Formulations

Integrable Equations and Motions of Plane Curves

The connection between motion of space or plane curves and integrable equations has drawn wide interest in the past and many results have been obtained. The pioneering work is due to Hasimoto where he showed in [1] that the nonlinear Schrödinger equation describes the motion of an isolated non-stretching thin vortex filament. Lamb [2] used the Hasimoto transformation to connect other motions of...

Integrable Non-qrt Mappings of the Plane

We construct 9-parameter and 13-parameter dynamical systems of the plane which map bi-quadratic curves to other bi-quadratic curves and return to the original curve after two iterations. These generalize the QRT maps which map each such curve to itself. The new families of maps include those that were found as reductions of integrable lattices by [11].

Rotation-minimizing osculating frames

An orthonormal frame (f1, f2, f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω · fi ≡ 0 — or, equivalently, the derivatives of fj and fk are both parallel to fi. The Frenet frame (t,p,b) along a space curve is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent ...

Osculating Random Walks on Cylinders

We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also consider families of such paths which together cover every edge of the...