Rotate a point according to a perpendicular point Let's say you have to points on a unit sphere (whose center is $O$), $X$ and $P$, such that $m \angle POX = 90^\circ = \frac \pi 2$. Additionally, we have an angle $\theta$. Imagine that we are viewing the sphere in such a way that $P$ is closest to us. We then rotate $X$ clockwise by $\theta$ to $X'$. Note that $m \angle POX' = 90^\circ$. Given $X$, $P$, and $\theta$, how do we find $X'$? Notes: * It would be nice if the calculation was also error correcting. By that I mean if $m \angle POX$ is not quite $90^\circ$ (say, $91^\circ$ or $89^\circ$), then $m \angle POX'$ is closer to $90^\circ$ then $m \angle POX$ is.

Let $\vec Y=\vec X\times\vec P$ and then $\vec {X'}=\cos\theta\cdot \vec X+\sin\theta\cdot \vec Y$. This gives the correct result for an original 90° angle, but for error correction, you may want to normalize $\vec{X'}$ (i.e., divide it by its length); this gives you an approximate result that is closer to 90° if the original angle was off.