Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to show this:

Given C(t) a unit-speed curve, using the usual Frenet-Serre frames T,N,B. Define the normal lines to C(t) to be the lines extending N, i.e, line segments containing N.

Then:

If all normal lines meet at a common point,

C(t) must be part of a circle.

Problem is that for curves, unlike for surfaces,

tangent plane is not defined , meaning that there

is no unique direction, no unique plane containing all

derivatives thru a point ; I know N is perpen-

dicular to T , by the first F-S equation, together

with the fact that <T,T>=1 (by unit speed; k is

curvature, assume k =/0):

T'=kN

<T,T>=1-> 2<T',T>=0

But there is a planeful worth of perpendiculars

to T', so I don't know how to tell which plane

contains N.

I tried to shift the axes so that the normals

meet at the origin, but I still cannot see it.

I thought I had found a counterexample, by

embedding a curve in S^2 , so that the tangent

space of the curve is a subspace of the tangent

space to S^2. Then I used the fact that the

normals to S^2 meet at the origin.

I showed this to a friend, who told me only

that my counterexample was worth 2 Kourics

(S.Park), and I am back to the drawing board.

Thanks for any ideas.

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# Frenet-Serre Frames and circles.

Can you offer guidance or do you also need help?

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