Differential Geometry Problems (2)

[septimus]

Homework Statement

1.

2.

Homework Equations

Frenet Formulas, definitions of curvature, torsion and generalized helix

The Attempt at a Solution

for 1)
I think I got part A down - I had α = λT + µN + νT, took the derivatives and plugged in the Frenet formulas to get:
λ′ − µκ − 1 = 0,
µ′ + λκ + ντ = 0
ν ′ − µτ = 0.
and i solved for τ and κ.

However, I'm having trouble with part B. I assume for part B, I should take the derivative of α = λT + µN + νT again, and use Frenet formulas to prove that this is equal to zero, but the algebra is not working out for me. Could anyone give me some hints or tips?

for 2)
I really don't know how to go about solving this problem; i was thinking of using this theorem

" a constant speed space curve p (t) is a generalized helix if and only if in a suitable orthogonal coordinate system the following holds
p(t)=q(t)+ct e'_3,
where q(t) is a constant speed curve in the x'y'-plane with curvature being nonzero everywhere, and c is a constant. Here e'_3 denotes the unit vector in the z'-direction."

but it doesn't seem to be getting me anywhere.



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ANY help would be really appreciated, thank you!

 

[septimus]

also curious: are we allowed to bump our posts?

thank you.