# How to determine the angles between a UAV and a sphere

I have an UAV modeled in three dimensions with let's say position coordinates $p_{uav} = (x_1,y_1,z_1)$ that is moving in a direction $d = (d_x,d_y,d_z)$ and a moving obstacle modeled as a sphere with known centre coordinates $p_{sph}=(x_2,y_2,z_2)$ and radius $r_{sph}$.

If I have a plane $p$ in the direction of movement of the UAV that intersects the sphere, I want to be able to calculate the angles with respect to the vehicle's movement formed by the tangents to the sphere in the plane $p$. In the figure, I would like to know how to calculate the angles $α_1$ and $α_2$.

If it helps, what I am looking is an extension in three dimensions for this:

Which is a vehicle in two dimensions ;it is obviously an easier problem which requires only the centre of the circle. However I am not really sure how to make it work in 3D, as supposedly the plane can intersect the sphere at any two points, not necessarily the centre.