Given an exact velocity and a "velocity range", what is the relative velocity range? I'm trying to calculate the relative velocity ($V_R$) between an exact velocity ($V_0$) and a velocity range ($V_1$).
The exact velocity ($V_0$) is represented simply by ($course$, $speed$).
The velocity range ($V_1$) is represented by a range of courses and a range of speeds, like so: $([course_{min}, course_{max}], [speed_{min}, speed_{max}])$ 
I'd like to obtain the relative velocity ($V_R = V_1 - V_0$), which would also be represented by $([course_{min}, course_{max}], [speed_{min}, speed_{max}])$.
Note that $(course, speed)$ is very similar to polar coordinates, with the only difference being that $course$ is zero when facing up (north) and increases clockwise.
I built a spreadsheet to see what patterns would emerge for different ranges of courses and speeds, and I came to the conclusion that the minimum and maximum relative courses & speeds often occur at the "corners" of course/speed space ($(course_{min}, speed_{min})$, $(course_{min}, speed_{max})$, $(course_{max}, speed_{min})$, $(course_{max}, speed_{min})$), but not always.
I'm wondering if there's a relatively simple equation to find $V_R$, or if I'll just need to perform a "brute force" calculation, where I loop through the courses and speeds (at some level of precision) and pick out the min/max values.
 A: To help advance the question a bit, I have made a graphic of what DanM has in mind. Say that $V_1$ should be in the following course/speed range: $([\pi/6,\pi/2],[1,2])$. This would give rise to the orange annular sector in the picture. But now, he wants to shift the values of this range by a vector $V_0$ with course/speed given by $(3\pi/4,\sqrt{2})$ (in Cartesian coordinates, this is just $(-1,1)$). Then, in what annular sector does the shifted orange sector fit? I have colored it blue in the picture.

In this case, the extreme values of the range in term of course/speed correspond to the corners of the orange range. But it is easy to construct a case where this is not so. Take the same orange sector, but now with $V_0$ having course/speed $(\pi/4,3)$. The picture becomes

The lower bound for the range of speeds is now determined by the circle boundary and not a corner. And to end, a picture of what happens when $V_0$ is just on the inner boundary:

To summarize, DanM's question is: when I change the origin of my coordinate system, in what minimal annular sector defined w.r.t. that new origin does the old sector, defined w.r.t. the old origin, belong?
