# Frenet-Serre Frames and circles.

1. Dec 17, 2007

### WWGD

Hi, everyone:

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.