prove the sufficient condition for orthogonal curve?
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I would like to prove that the following two statements are necessary and sufficient conditions that a curve is a helix. I know that a helix is a space curve with the property that the tangent to the curve at every point makes a constant angle with a fixed direction. I am attempting to prove it but am not sure if some of the logic is right.
(i) The Principal normal is orthogonal to a fixed direction
(ii) κω=c where κ is the curvature, ω is the torsion, and c is a constant.
Now for (i) we prove the necessary condition. If my space curve is a helix, it has parametric equations x=acosθ, y=asinθ, z=bθ. Assume without loss of generality that a=1 and b=1. Then performing calculations I find that the principal normal n→=⟨−cosθ,−sinθ,0⟩ which is orthogonal to any vector of the form ⟨0,0,a⟩, a some constant.
For the sufficient condition, suppose n→⋅a→=0, a→ some vector with fixed direction. Then by the frenet- serret formulas,
dT→ds=κn→, and hence taking the dot product with a→ on both sides gives the equation dT→ds⋅a→=0.
(∗) Now here's the logic. If T′(s) is perpendicular to a→, where T is my unit tangent vector, then since T is perpendicular to T′(s) it follows that T itself makes a constant angle with a→. Is this bit of logic right??
Now for part (ii), the bit on necessity is easy as I just use the same helix and get that κω=1. It is just the bit on sufficiency that is difficult. I have no idea how to go from κω=c to the fact the tangent vector makes a constant angle to a fixed direction.
Please do not give me any full answers for the last bit, but instead pose me some questions that my motivate my understanding of the problem.
Edit: Proof of the neccesary condition for (i). If T⋅e¯=c, e¯ some vector with fixed direction and c a constant, then differentiating both sides you get T′(s)⋅e¯=κn¯⋅e¯=0, by the first of the frenet serret formulas which means that the unit normal vector is perpendicular to some vector with fixed direction.