Question

Prove that the ratio of the curvature to the torsion is constant at all pts

Answer 1

Prove that if the tangent lines of a curve make a constant angle with a fixed direction, then the ratio of its curvature to its torsion is constant.

So, I started by letting the curve be parameterized by arclength for convenience. Then, I let the fixed direction be the principal normal of the curve (as suggested by my professor). I know that the ratio of curvature to torsion is constant for a helix, so I was thinking of trying to prove that the assumptions imply that the curve must be a helix.

I tried using the cosine similarity formula as follows (with TT being the tangent vector, and uubeing my fixed principal normal direction:

cos(θ)=T⋅u∥T∥∥u∥cos(θ)=T⋅u‖T‖‖u‖ is constant

I think I can say that both TT and uu are unit, so then I'd have that cos(θ)=T⋅ucos(θ)=T⋅u is constant.

Then, I was thinking if I showed dds(T⋅u)=0dds(T⋅u)=0, then I could somehow relate that back to curvature.

Am I on the right track?

Thank you very much for any help!