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Math Help - Normal vector of spherical curve

  1. #1
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    Normal vector of spherical curve

    I just started a differential geometry course and I am very far removed from a lot of fundamental relevant background material (such as basic calculus and analysis) so please bear with me.

    True or False: For any spherical curve,  c(s) , the center of the osculating sphere, denoted  m(s) , is collinear with the normal vector.

    First things first, I'm pretty sure this is true. On a spherical curve, the osculating sphere is constant for all  s . So, in particular, the center is constant. I think that the normal vector, N(s), points directly at the center of the osculating sphere (in fact, that the terminal point of the normal vector IS the center, at any point along the curve)...well either that or -N(s) does.

    Does this sound like a decent approach?
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  2. #2
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    I got that, but what about the converse? If \exists  p , a fixed point that lies in every normal plane of a curve c(s)   , then the curve must be spherical.

    Here's my thinking so far: A curve is spherical if and only if  \frac{\tau (s)}{\kappa (s)}-\frac{d}{ds}(\frac{\kappa '(s)}{\tau (s) \kappa (s) ^2})=0 . I would like to use the assumption that there is a fixed point in every normal plane to reach this equality, but I'm pretty lost.

    I'm not clear on what it means for p to be in every normal plane of c(s)   . Does this mean that  \forall s \exists \alpha ,\, \beta  such that \alpha \overrightarrow{N}(s) + \beta \overrightarrow{B}(s) = p   ? But if this is so, I don't technically get a point I get a vector from the origin to the point p (I guess this probably isn't critical, but I still feel uneasy somewhat).

    If I'm on the right track, I still don't know how to tie these things together. If I'm not on the right track, could someone help me find where to tap my cane?
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