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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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    It's not really clear to me what you are trying to model. Do these velocities change with time? Is your "exact" velocity the constant, *average velocity*? Is your velocity "range" $V_1$ the *instantaneous* velocity at time $t$? Is there a constraint on these aside from that the object they model starts at some time $t_0$ and position $x_0$ (where $x$ is distance along a line or arc length along a curve) and ends at some time $t_1$ and position $x_1$? If this is your model, then $x=x(t)$ is a function of $t$ and $$v_\text{avg}=\frac{x_1-x_0}{t_1-t_0},\qquad v_\text{inst}=\frac{dx}{dt}$$.2012-04-01
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    Could you show the rest of your work as I'm not really sure what you are asking. Presumably, you know how to compute $V_R$ given one vector $V_1$, yes? So why is it a problem to determine the range of possible values?2012-04-01
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    Usually, we assume that $\frac{dx}{dt}$ is continuous, but it can even approach infinity at a point. For example, $x(t)=\sqrt{t}$ for $t\in[0,1]$ has average velocity $1$ but $x'(0)=\infty$.2012-04-01
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    @bgins: I think that is irrelevant to OP's question. His is purely a trigonometry question. But I don't see why he is stuck on it, as he seems to be able to compute things.2012-04-01
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    Actually, I've just noticed your other question about the annular sector, and I realize now what you expect. But that is not possible. As you can see, if $V_1$ is part of such a sector, subtracting $V_0$ will just translate that sector. But since polar coordinates are always described w.r.t. the same fixed origin, the expression for the translated domain in terms of polar coordinates will become unwieldy. You will not be able to put it in the form you desire.2012-04-01
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    @bgins, the velocities can change over time, but I only need to calculate instantaneous values periodically (say, every second). The "velocity range" is intended to model an estimate based on partial data where the exact course and speed are not known, but the range of these values is known.2012-04-01
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    @Raskolnikov, yes, good catch, I'm trying to visualize course and speed uncertainty as an annular sector. I don't want to translate the sector, I want to calculate a new sector. I only care about the bounds of the sector (the min/max values), and I'm simply looking for an equation to calculate these bounds. Does it make sense now?2012-04-01
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    Yes, but that means you also don't mind including vectors that aren't in your original sector. Obviously, the new sector will always be much larger than the original one, introducing in a sense artificial uncertainty.2012-04-01
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    Is the true course constant? Is the starting position (say, the origin) and time (say, $t=0$) known? If so, the true position is $(x,y)=(s(t)\sin\theta,s(t)\cos\theta)$ where $s(t)=\int_0^tv(\tau)d\tau$ & $v(t)=s'(t)$. So you are given estimates $(\theta_n,v_n)$ at times $t_n=n\Delta t$ according to some error model. You could call these random variables $\Theta_n,V_n$, which might be easier to derive from $X_n,Y_n$ depending on your error model. Do you want to transform an error model defined in Cartesian coordinates, say a 2D Gaussian, into polar coordinates? Or Iterative Bayesian updates?2012-04-01
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    @bgins, you may be misunderstanding my question. I'm not trying to predict future locations/velocities, I'm just trying to figure out what the instantaneous velocity would be for true versus relative. That would normally be easy (just subtract $V_1 - V_0$), but it's difficult in this case because the components of $V_1$ are ranges not exact values. So, time should not be a factor in this.2012-04-01
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    I admit, I don't quite understand what you want. That's why I'm trying to help you formulate an equation that everyone can relate to.2012-04-01
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    @Raskolnikov, 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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