Geodesic
The Evolution of the Local Induction Approximation for a Regular Polygon
Francisco de la Hoz1 ; Luis Vega2
1 Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
2 Department of Mathematics, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain. BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
ESAIM. Proceedings, Tome 45 (2014), pp. 447-455.

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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.
DOI : 10.1051/proc/201445046

Francisco de la Hoz 1 ; Luis Vega 2

1 Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
2 Department of Mathematics, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain. BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain 
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     author = {Francisco de la Hoz and Luis Vega},
     title = {The {Evolution} of the {Local} {Induction} {Approximation} for a {Regular} {Polygon}},
     journal = {ESAIM. Proceedings},
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     doi = {10.1051/proc/201445046},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201445046/}
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Francisco de la Hoz; Luis Vega. The Evolution of the Local Induction Approximation for a Regular Polygon. ESAIM. Proceedings, Tome 45 (2014), pp. 447-455. doi : 10.1051/proc/201445046. http://geodesic.mathdoc.fr/articles/10.1051/proc/201445046/

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