The two purple spheres have known coordinates, and can be anywhere in 3d space. The start of the darker grey part is always the highest point of the cylinder. The radius of the cylinder is also known.
Now I can calculate the position of the black dot if the angle of the darker grey is 90 degrees (or 1/2 pi rad) using the cross product.
How can I calculate the position if the angle is different?
In this example the coordinates are:
( 3, 4, 3)
(13,12, 8)
Radius = 2
Angle = 117 degrees
You need to solve equation $\vec t\times\vec x=\frac{r^2sin\alpha}{|\vec{S1S2}|}\vec{S1S2}$ where $\vec t$ is the top vector
In your example $\vec{S1S2}=(10,8,5)$ $\vec t=k(10,8,-164/5)$ $k=-2/|(10,8,-164/5)|$ (if z is the blue axis)