Shortest distance between two different helix lines

Question

I have two helix lines with center distance from each other. The lines had different rotational directions and radii and I'm interested to find a mathematical way to find the shortest distance between them. The lines can be described as follows:

Helix 1: \begin{align} x_1 &= R_1\cos (u) \\ y_1 &= R_1\sin (u) \\ z_1 &= c_1 u \end{align}

Helix 2: \begin{align} x_2 &= R_2\cos (v) + \text{CenterDistance} \\ y_2 &= R_2\sin (v) \\ z_2 &= c_2 v \end{align}

Answer 1

The projections to the $xy$-plane are circles of radius $R_1$ and $R_2$ respectively. Hence the distance is certainly at least $|R_1-R_2|$. But for $u=0$, we verify that $|R_1-R_2|$ is actually achieved.