I know that the parametrization of a helix centered along the z-axis is $\vec r(t)=<cos(t),sin(t),t>$. However, is there some way to center the helix along some arbitrary vector $\vec n$? I was thinking maybe I could use spherical coordinates and then $\rho=t\hat n$, but I can't figure out how to solve for $\phi$ and $\theta$. Any help is appreciated. Thank you.
Centering a Helix along an Arbitrary Vector in 3D
geometryspherical coordinatesvectors
Best Answer
I would find $\mathbf u,\mathbf v$ such that $\mathbf u,\mathbf v$ are unit vectors perpendicular to $\mathbf n$
For any $\mathbf n$ it is pretty easy to find one orthogonal vector. Then take the cross product to find the other. Scale them down to unit vectors.
$\mathbf n t + \mathbf u \cos t + \mathbf v \sin t$ would be your helix.