The discriminant of a certain quadratic equation which determines when two circles (hyperspheres?) will collide, provided they travel with a constant velocity is
$$(\Delta v\cdot\Delta s)^2-(\Delta v\cdot\Delta v)\times(\Delta s\cdot\Delta s-r^2)$$
where $\Delta v$ is the difference in the velocities of two circles, $\Delta s$ is the difference of their positions, and $r$ is the sum of their radii. What is the minimal rotation of $\Delta v$ that will make this discriminant negative, and therefore avoid the collision of the circles?
I have spent many, many hours trying to massage the above into various mathematical packages, and I've received many incomprehensible multi-screen equations and many useless results with $cos(\theta)$ and $sin(\theta)$ on both sides, which I could not resolve.
Any suggestions would be nice, a numerical solution would also suffice.