Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.
What is your definition of a helix? I've never heard that as a technical term before.
A helix is a curve in 3-dimensional space. The following parametrization in Cartesian coordinates defines a helix:
x(t) = cos(t),
y(t) = sin(t),
z(t) = t.
OK, then what is
? and what do you mean by
I suspect dammahom59
, the curvature and torsion. In which case, it follows immediately from the Frenet-Serret formula and uniqueness of solutions, remembering a characterisation of helix is a space curve with a constant direction
orthogonal to the principal normal vector (exercise: prove this characterisation from the usual definition that a helix is a space curve where the tangent vector makes a constant angle to a fixed direction).
Yes, that is exactly what I meant, thank you.