Diff Geo - Ferret Serret Apparatus

[sparkster]

The problem says this:

Let a(s) be a C^k curve in the x,y plane. Prove that if the curvature /=0 then the torsion is identically 0.

It gives a hint to note that there exist diff functions x(s), y(s) such that a(s)=(x(s), y(s), 0). Then show that B (Binormal vector field) = +/- (0,0,1).

I can get that the x and y coordinates of B are 0, but not the 1 part.

Also, I don't follow the reasoning behind the problem at all. If the curvature is zero, then you cannot calculate the normal, binormal, or torsion (at least not with the definitions we were given for them). So the problem seems to be saying that if the torsion exists, it must be 0!

But this can't be the case. An example in the same chapter of the text gives a(s)=r cos(ws), r sin (ws), hws). The curvature turns out to be non-zero, but the torsion is non-zero too. What am I not following?

 

[StatusX]

But this can't be the case. An example in the same chapter of the text gives a(s)=r cos(ws), r sin (ws), hws). The curvature turns out to be non-zero, but the torsion is non-zero too. What am I not following?

But that curve doesn't lie in the xy plane, as this problem requires.

 

[sparkster]

But that curve doesn't lie in the xy plane, as this problem requires.

D'oh!

Okay, so in the xy plane, any curve with curvature zero, doesn't have torsion, etc. That makes sense. Now I have to figure out where I'm going wrong in the proof.