Rotation axis in axis-angle/Rodrigues' rotation formula?

Question

I am trying to understand the axis-angle/Rodrigues' rotation formula with https://en.wikipedia.org/w/index.php?title=Rodrigues%27_rotation_formula. As the article says, I can obtain my rotation axis $k$ by cross-product multiplying two intersecting vectors $a$ and $b$ that define a plane.

My confusion is that the so obtained $k$ is just a directional vector, not rooted at any particular point in the coordinate system. But to serve as a rotation axis, the vector has to be rooted at a certain point. Say, I compute $a\times b=k=(0, 0, 1)$. If I root the rotation axis at different starting points, the rotations are different, although the directional vector $k$ is still $(0, 0, 1)$.

So in a nutshell, I don't understand how the rotation axis is rooted instead of being merely directional.

Answer 1

The formulas shown in the Wikipedia article for Rodrigues' rotation formula are for rotation around an axis through the origin.

In order to rotate around some other axis, first perform a translation $T$ that maps some point on your desired rotation axis to the origin; then perform a rotation by the desired angle around the axis through the origin parallel to your desired rotation axis; then perform the inverse of the translation $T.$