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 .

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- Nov 5th 2008, 02:58 PMdori1123parabolic 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 . - Jun 8th 2009, 08:52 PMRebesques
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).