Thread: parabolic or planar points

Show that if a surface is tangent to a plane along a curve, then the points of this curve are either parabolic or planar.

Defition: A point of a surface is parabolic if with =/= . A point of a surface is planar if .

Let the surface be parametrized by and the curve by .
Since is also a plane curve, we have that the binormal vector satisfies and that (as there is no torsion). So for points along the curve . Now this gives are linearly dependent at , so if they are not zero (and the point planar) then (and so the point is parabolic).