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Zh. Mat. Fiz. Anal. Geom., 2009, Volume 5, Number 1, Pages 12–24 (Mi jmag114)  

Retroreflecting curves in nonstandard analysis

R. Almeidaa, V. Nevesa, A. Plakhovab

a Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
b Institute of Mathematical and Physical Sciences, University of Aberystwyth, Aberystwyth SY23 3BZ, Ceredigion, UK

Abstract: We present a direct construction of retroreecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class $C^1$, except for a hyper-nite set of values, such that the probability of a particle being reected from the curve with the velocity opposite to the velocity of incidence, is innitely close to 1. The constructed curves are of two kinds: a curve innitely close to a straight line and a curve innitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: nd the curve of maximum resistance innitely close to a given curve.

Key words and phrases: Nonstandard Analysis, retroreflectors, maximum resistance problems, reflection, billiards.

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Bibliographic databases:
MSC: 26E35, 49K30, 49Q10
Received: 29.03.2008
Language:

Citation: R. Almeida, V. Neves, A. Plakhov, “Retroreflecting curves in nonstandard analysis”, Zh. Mat. Fiz. Anal. Geom., 5:1 (2009), 12–24

Citation in format AMSBIB
\Bibitem{AlmNevPla09}
\by R.~Almeida, V.~Neves, A.~Plakhov
\paper Retroreflecting curves in nonstandard analysis
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2009
\vol 5
\issue 1
\pages 12--24
\mathnet{http://mi.mathnet.ru/jmag114}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2528397}
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\elib{http://elibrary.ru/item.asp?id=12796277}


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