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...
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EDP Sciences
2014-09-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | http://dx.doi.org/10.1051/proc/201445046 |
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doaj-9f1f354ba319476aadff73dda63a32382021-07-15T14:07:21ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592014-09-014544745510.1051/proc/201445046proc144546The Evolution of the Local Induction Approximation for a Regular Polygon *de la Hoz Francisco0Vega Luis1Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHUDepartment of Mathematics, Faculty of Science and Technology, University of the Basque Country UPV/EHUIn 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.http://dx.doi.org/10.1051/proc/201445046 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
de la Hoz Francisco Vega Luis |
| spellingShingle |
de la Hoz Francisco Vega Luis The Evolution of the Local Induction Approximation for a Regular Polygon * ESAIM: Proceedings and Surveys |
| author_facet |
de la Hoz Francisco Vega Luis |
| author_sort |
de la Hoz Francisco |
| title |
The Evolution of the Local Induction Approximation for a Regular Polygon * |
| title_short |
The Evolution of the Local Induction Approximation for a Regular Polygon * |
| title_full |
The Evolution of the Local Induction Approximation for a Regular Polygon * |
| title_fullStr |
The Evolution of the Local Induction Approximation for a Regular Polygon * |
| title_full_unstemmed |
The Evolution of the Local Induction Approximation for a Regular Polygon * |
| title_sort |
evolution of the local induction approximation for a regular polygon * |
| publisher |
EDP Sciences |
| series |
ESAIM: Proceedings and Surveys |
| issn |
2267-3059 |
| publishDate |
2014-09-01 |
| description |
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. |
| url |
http://dx.doi.org/10.1051/proc/201445046 |
| work_keys_str_mv |
AT delahozfrancisco theevolutionofthelocalinductionapproximationforaregularpolygon AT vegaluis theevolutionofthelocalinductionapproximationforaregularpolygon AT delahozfrancisco evolutionofthelocalinductionapproximationforaregularpolygon AT vegaluis evolutionofthelocalinductionapproximationforaregularpolygon |
| _version_ |
1721300275120046080 |