The Evolution of the Local Induction Approximation for a Regular Polygon *

Abstract

In this paper, we consider the so-called local induction approximation (LIA): $$ \Xt = \Xs\wedge\Xss, $$ X t = X s ∧ X ss , where ∧ is the usual cross product, and s denotes the arc-length parametrization. We study its evolution, taking planar regular polygons of M sides as initial data. Assuming uniqueness and bearing in mind the invariances and symmetries of the problem, we are able to fully characterize, by algebraic means, X(s,t) and its derivative, the tangent vector T(s,t), at times t which are rational multiples of 2π/M2. We show that the values at those instants are intimately related to the generalized quadratic Gauß sums.