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Show that if a 3-D surface is tangent to a plane along a curve, then the points of this curve are either parabolic or planar.

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    Here is a reference: Differential Geometry of Curves and Surfaces [Manfredo P. do carmo] section 3-2 exercise 2.2013-11-30

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For every point of the curve, the surfaces curvature in the direction tangent to the curve is zero. Therefor the point is parabolic or planar

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    I mean "curvature along a direction". Cfr. Shape operator and the principal directions of curvature.2013-11-30