The formula for the point position rotated around an axis of a sphere

Question

Please let me know the formula for the point position rotated around an axis of a sphere.

In detail, I want to do as follows. Given:

any point $p_1$ to decide the rotation axis ax of a sphere of ( radius r and center c);
any point $p_2$ rotated around ax;
any angle $angle$ as the measure of the rotation of the sphere around ax,

then compute the new position of $p_2$ by a formula.

Yes , I want the center c is at the origin and the rotation axis goes through p1 and c .

I found a rotation matrix formula at the bottom in the wikipedia page of "Rodrigues rotation formula ". Is it only for the special case as , the rotation axis goes through the origin?

Thank you very much.

Are you assuming the center of the sphere is at the origin, and the axis of rotatioon will go through the origin and $p_1$? — Arturo Magidin, Jun 13 '11 at 05:53
Yes , I want the center c is at the origin and the rotation axis goes through p1 and c . — seven_swodniw, Jun 13 '11 at 06:11

John Alexiou · Accepted Answer · 2011-06-13T06:59:29.857

3

Use the Rodrigues rotation formula ${\rm Rot}[\vec{k},\theta,\vec{v}]$ for arbitrary axis.

$$\vec{p}_2 = {\rm Rot}[\vec{p}_1-\vec{c}, \theta,\vec{p}_2-\vec{c}] + \vec{c}$$

where $\theta$ is the angle.

edited Jun 13 '11 at 06:59

answered Jun 13 '11 at 06:44

John Alexiou

13,816

Thank you very much for the answer. – seven_swodniw Jun 13 '11 at 11:25
I found a rotation matrix formula at the bottom in the wikipedia page of "Rodrigues rotation formula ". Is it only for the special case as , the rotation axis goes through the origin? – seven_swodniw Jun 13 '11 at 17:59
Thats why I subtract $\vec{c}$ and then add it again later. – John Alexiou Jun 16 '11 at 14:25

The formula for the point position rotated around an axis of a sphere

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