Calculate 3d point on sphere's surface given a cone inside it

Question

I have a sphere with its center at $(0,0,0)$ and 2 vectors $\vec a$ and $\vec b$ and both of them have a length of spheres radius

$\Vert \vec a \Vert = \Vert \vec b \Vert =R$

Angle between these vectors ($\alpha$) is also known.

If $\vec a$ circles around the $\vec b$ keeping the angle $\alpha$ unchanged we are kind of getting a cone inside the sphere like in this picture.

How can I calculate the coordinates of $\vec a$ if it would rotate $\beta$ degrees around the $\vec b$ ?

Answer 1

To make thinks simpler lets assume that this is a unit sphere.

$p = (a\cdot b) b$ is the projection of $a$ onto $b$

$v = a - (a\cdot b) b$ is orthogonal to $b$

$p+v = a$

$u = b\times a$ is orthogonal to both $v$ and $b$

$\|u\| = \|v\|$

the circle: $p+v\cos t + u\sin t$