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Thread: parabolic or planar points

  1. #1
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    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 $\displaystyle det(dN_p) = 0$ with $\displaystyle dN_p$ =/= $\displaystyle 0$. A point of a surface is planar if $\displaystyle dN_p = 0$.
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    Super Member Rebesques's Avatar
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    Let the surface be parametrized by $\displaystyle F(u,v), \ (u,v)\in D$ and the curve by $\displaystyle x(t)=F(u(t),v(t)), \ t\in I$.
    Since $\displaystyle x$ is also a plane curve, we have that the binormal vector $\displaystyle b(t)$ satisfies $\displaystyle b(t)=\pm N(u(t),v(t))$ and that $\displaystyle b'=0$ (as there is no torsion). So for points $\displaystyle p$ along the curve $\displaystyle 0=b'=N_uu'+N_vv'={\rm d}N_p(u',v')=0$. Now this gives$\displaystyle \{N_u,N_v\}$ are linearly dependent at $\displaystyle p$, so if they are not zero (and the point $\displaystyle p$ planar) then $\displaystyle {\rm det}[{\rm d}N_p]=0$ (and so the point is parabolic).
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