  We study the normal map for plane projective curves, i.e., the map
associating to every regular point of the curve the normal line at the point in
the dual space. We first observe that the normal map is always birational and
then we use this fact to show that for smooth curves of degree higher than four
the normal map uniquely determines the curve. Our proof works in characteristic
zero and in positive characteristic higher than the degree of the curve. We
notice also that in high characteristic strange curves provide examples of
different plane curves with same curve of normal lines. We will reinterpret our
results also in the modern terminology of bottlenecks of algebraic curves.
