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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.

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    @Raskol$n$i$k$ov, I've been playing around with this using brute-force calculations, and it does seem that there are some combinations that yield very large relative course ranges. This seems to happen when the relative speed approaches zero. I decided to deal with this by calculating the relative course using the midpoint values, then ignoring any course error that differs by more than 90 degrees from the midpoint course. At this point, I think using brute force is probably the best I'm going to get.2012-04-02

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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.

Annular sector 1

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

Annular sector 2

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:

Annular sector 3

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?