  B\"acklund transformations for smooth and ``space discrete''
  Hashimoto surfaces are discussed and a geometric interpretation is given. It
is shown that the complex curvature of a discrete space curve evolves with the
discrete nonlinear Schr\"odinger equation (NLSE) of Ablowitz and Ladik, when
the curve evolves with the Hashimoto or smoke ring flow. A doubly discrete
Hashimoto flow is derived and it is shown, that in this case the complex
curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete
NLSE. Elastic curves (curves that evolve by rigid motion only under the
Hashimoto flow) in the discrete and doubly discrete case are shown to be the
same. There is an online version of this paper, that can be viewed using any
recent web browser that has JAVA support enabled. It includes two additional
java applets. It can be found at
http://www-sfb288.math.tu-berlin.de/Publications/online/smokeringsOnline/
