There are two objects A and B that can be described in position space and velocity space. The position space describes the instantaneous positions of the objects while the velocity space describes the rate of change in position of the objects. They are both of uniform density in both position and velocity space. For each individual object the location of one of its points in position space is unrelated to that point location in velocity space and vice verse. Both objects are perfect spheres in both position and velocity space.
The center of object A in position space has coordinates ($X_A$, $Y_A$, $Z_A$), and the center of object A in velocity space has coordinates ($v_{X_A}$, $v_{Y_A}$, $v_{Z_A}$).
The center of object B in position space has coordinates ($X_B$, $Y_B$, $Z_B$), and the center of object B in velocity space has coordinates ($v_{X_B}$, $v_{Y_B}$, $v_{Z_B}$).
Object A has a volume of $V_A$ in position space and a volume of $V_{v_A}$ in velocity space. Object B has a volume of $V_B$ in position space and a volume of $V_{v_B}$ in velocity space.
Now we can take pairs of points from the two objects, in which each point has coordinates in position space and coordinates in velocity space, and in which one point is from object A and one point is from object B. These pairs of points have a displacement between the points in position space and a velocity between the points in velocity space. My first question is how do I calculate the fraction of pairs of points, in which the displacement between the points in position space is within 1cm of being in the same direction as the velocity between the points in velocity space? My second question is how do I calculate the fraction of pairs of points, in which the velocity between the points in velocity space is within 1cm/s of being in the same direction as the displacement between the points in position space?
The only sphere in 3-space whose image in velocity space at some point "has uniform density" in the sense described in the answers to comments above...is in purely translational motion, not rotation.
The "direction of the distance in position space" is presumably a unit vector, computed by taking $\frac{v}{\|v\|}$ for $v$ the displacement vector. The same goes for the direction of the distance in velocity space. Unfortunately, unit vectors don't come with units like "cm", so the first question doesn't really make sense. (And nor does the second, of course).
It appears to me that this question needs a little bit of refinement.