Helices , Hasimoto Surfaces and B

Travelling wave solutions to the vortex lament ow generated by elastica produce surfaces in R 3 that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classiied in the special cases where the helices are all congruent or are all generated by a single screw motion. The rst case yields a new characterization for the BB acklund transformation for constant torsion curves in R 3 , previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in H 3 or S 3 leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.