Retroreflecting curves in nonstandard analysis

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.