Set of rotations necessary to connect two points in R³ using a thin cylinder

Question

I have been scratching my head for days trying to answer this question.

Suppose i have 2 points on three-dimensional space, say, $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, and they are separated by a distance $d$. Also, imagine i have a very thin cylinder, whose height is also $d$ (for this purpose, it may as well be thought of as a line). His center of mass is located exactly on point $A$. My question is: how do i go about calculating the offsets in the cylinder's position (supposing that "the cylinder's position is $(a, b, c) $" means "the cylinder's center of mass, assuming constant density, is located at $(a, b, c)$"), as well as (most importantly) the set of rotations necessary so that one end of the cylinder is located at $A$ and the other end is located at $B$?

I say the rotations are the most important, because i suppose it is clear that the cylinder's center of mass should be positioned exactly between $A$ and $B$, so it's center of mass position should be $(\frac{x_2 - x_1}{2}, \frac{y_2 - y_1}{2}, \frac{z_2 - z_1}{2})$. But how to calculate the set of rotations around the $x, y$ and $z$ axis?

I suppose quaternions should help me on that, but i have never studied them. Is there another way around?

Thanks in advance, folks!

Answer 1

The initial orientation of the cylinder is of some significance. But whatever it is, the following sequence of rotations works:

1. rotate about the $z$ axis until the cylinder is in the $xz$-plane
2. rotate about the $y$ axis until the $z$-coordinate of one end is what it should be, namely $(z_2+z_1)/2$
3. rotate about the $z$ axis until the aforementioned end is at the midpoint of the segment $AB$.

After these rotations, the translation that you described bring the cylinder to the desired position.