The Evolution of the Local Induction Approximation for a Regular Polygon *
In this paper, we consider the so-called local induction approximation (LIA): $$ \Xt = \Xs\wedge\Xss, $$ X t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
EDP Sciences
2014-09-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | http://dx.doi.org/10.1051/proc/201445046 |
| Summary: | 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. |
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| ISSN: | 2267-3059 |