Frenet-Serre Frames and circles.

[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.

 

[jvicens]

Hello,

First of all, I want to clarify that the tangent plane is defined for curves, it just may not be unique. The tangent plane at a point on a curve is the plane that contains the tangent vector and all of its derivatives at that point. So while there may be a "planeful" of perpendiculars to the tangent vector, there is only one tangent plane.

Now, let's consider the problem at hand. If all normal lines meet at a common point, then we can assume that this point is the center of the circle. This is because, for a circle, all normal lines will be perpendicular to the radius at that point. This means that the tangent vector at that point will also be perpendicular to the radius, and therefore will lie in the tangent plane. Since all normal lines meet at this point, it follows that all tangent vectors at each point on the curve will also lie in this same tangent plane. This indicates that the curve is part of a circle.

To further prove this, we can use the second Frenet-Serre equation, which states that N' = -kT. This means that the normal vector is always perpendicular to the tangent vector, and therefore also perpendicular to the radius at each point on the curve. This confirms that the curve is indeed part of a circle.

I hope this helps to clarify the problem. Let me know if you have any further questions or concerns.

 

[tacman]

Hello,

Thank you for sharing your thoughts on Frenet-Serre frames and circles. I can see that you have put a lot of effort into trying to understand this concept. Let me try to provide some insights that might help you in your exploration.

Firstly, you are correct in stating that for curves, unlike surfaces, the tangent plane is not uniquely defined. This is because a curve is a one-dimensional object, while a surface is a two-dimensional object. Therefore, there are multiple planes that can contain the tangent vector at a given point on a curve.

However, in the case of Frenet-Serre frames, we are not concerned with the unique tangent plane. Instead, we are interested in the normal line, which is defined as the line perpendicular to the tangent vector at a given point on the curve. This normal line is unique and can be found by extending the normal vector N from the Frenet-Serre frame.

Now, coming to your question about how to tell which plane contains N, let's consider the definition of the normal vector N. It is defined as the cross product of the tangent vector T and the binormal vector B. Therefore, the plane containing N is perpendicular to both T and B. This means that if you shift the axes so that the normals meet at the origin, the plane containing N will also pass through the origin.

To visualize this, you can imagine a circle on a flat piece of paper. If you draw a tangent vector at any point on the circle and extend it, it will give you the tangent line. Now, if you draw the normal vector at the same point and extend it, it will give you the normal line, which will be perpendicular to the tangent line. This same concept applies to curves in three-dimensional space as well.

I hope this explanation helps you in understanding Frenet-Serre frames and their relationship to circles. Keep exploring and don't give up, as mathematics can be challenging but also very rewarding. All the best!