Trilateration/Measurement of a Separation From A Distance

[chrisa88]

Homework Statement

This was done on a sidewalk. Two boxes were set up a distance apart with one line on each box. We measured the distance of ten points going out from the boxes (ten cement tiles on the sidewalk) and were able to use a meter stick to measure by eye from each ten points the separation between the two lines on the boxes. So what I know is the separation I measured (from afar) using the meter stick, the length of my outstretched arms, and the distance each measurement was taken from the boxes. Attached is a picture of the triangle and variables drawn out.
h_o= separation between two lines on boxes
d= distance from the point of the measurement to the boxes
x= length of my outstretched arms holding the meter stick
y= measurement taken by eye from the meter stick.

Homework Equations

I was given this equation h/d=y/x
but when I plug this in my distance between the two lines on the boxes continual increases when the distance away from the boxes increases, which is intuitive based on the equation.


3. Question
I know about triangulation, but this does not seem to deal with triangulation since we are not using angles, rather it seems to deal with trilateration, which I have not been able to find any useful information on google or anywhere. Could someone please help me figure out how to do this? My lab report is due tomorrow at 10:20AM and I have not been able to start since I do not understand the basis of measuring the distance between the two boxes using our knowns.

Please!
Thank you,

Christopher Anderson

 

[chrisa88]

OK I'm pretty sure I figured out how to do this quite a while ago (I've been questioning myself because the fact that the measurement of the separation increases as the distances away from the two lines increases), but the thing that is stumping me is the fact that the distance between the two lines increases with the distance away from the boxes so how am I supposed to come to a conclusion about what the real measurement of the separation is when this occurs?? We did do two trials, one with a larger separation and one with a smaller separation, but using the same distances away from the boxes.

Does anyone understand how I'm supposed to come up with a final measurement of the separation? I'm wondering if there is some ratio I'm supposed to use between the two trials to come up with the actual distance between the two lines.

Any help or thoughts are appreciated..

 

[Simon Bridge]

Hint: you can get the apex angle via trig.

If you have the angle subtended by a known separation a known distance away, then you can figure out other separations. i.e. if 1m subtends 1 degree, and your unknown separation subtends 3 degrees ... what is the unknown separation?

 

[chrisa88]

Hint: you can get the apex angle via trig.

If you have the angle subtended by a known separation a known distance away, then you can figure out other separations. i.e. if 1m subtends 1 degree, and your unknown separation subtends 3 degrees ... what is the unknown separation?

Thank you, but I believe I have figured this out several different ways. One way is the congruence of right triangles which can then be multiplied by two (concerning the measurement of the gap) to determine the length of the total, or using the h_o/d=y/x method as given to me by my teacher. However, my issue now is how to discover the actual measurement since, for instance, the calculated measurement of the gap increases as my distances, from ten different measuring spots, away from the boxes increases and the measured gap decreases (measured on the meter stick; since getting further away two points appear to be closer). As I said in my previous reply to myself, we did two trials and I'm assuming I'm supposed to make some ratio between the two trials to come up with the actual calculated gap measurement, but I'm not exactly sure how to do this. Any idea?
Thank you!

 

[Simon Bridge]

Did I misunderstand - you did a trial to calibrate the measurements and another trial to measure an unknown length?

 

[chrisa88]

No we did two trials with different gaps between the lines to be measured. The gap length for both trials was unknown. For trial one the gap length was larger than trial two, the only measurements that stay the same are the distances of each measurement location and the length of my arms outstretched holding the meter stick. From this we are supposed to be able to measure the length of the gap between the two lines (red lines in picture) on the boxes. The calculation he discussed in class was h_o/d=y/x, but my issue is since the length of my arms, variable x, is not changing as the measurements are being taken further and further away, the calculated gap length is not becoming evident as it should since h_o=(y*d)/x, since x stays the same. So how am I supposed to come to a final gap length since the numerator increases but the denominator stays the same? A more detailed picture is included and yet again the only difference in trials is the h_o length decreased for trial two (they moved the boxes closer together).

 

[Simon Bridge]

Well in that case there would be nothing to gain taking ratios between the two trials.

 

[chrisa88]

That is exactly what I was thinking! Am I correct in saying that the only way to come to a conclusive calculation of the gap length over all ten measurements (taken ten different distances away from the boxes) is if the distance x changed as the total distance away from the boxes changed? From my understanding since x does not change the gap length is not going to be conclusive because the numerator of the ratio just gets larger as the denominator , x, stays the same? Can you see any way to make this calculation using just the apparent measurement of the gap length on the meter stick with the distance away from the boxes?

 

[Simon Bridge]

OK - so you know x and d, you measure y, and so deduce h - which is an unknown separation of the boxes.

You do this for N+1 different distances - $\{d_0,d_1,d_2,\cdots ,d_N\}$
So you would keep x the same for each measurement, and collect $\{y_0,y_1,y_2,\cdots ,y_N\}$ corresponding to the different distances. So the nth unknown distance is found from: $$h_n = \frac{x}{y_n}d_n$$ ... as d gets bigger, y gets smaller.

Is h different for the different distances too - or do you leave the boxes with the same separation?
Do you increase the distance by a fixed interval, so the nth distance is $d_n = d_{n-1}+\delta = d_0+n\delta$ ... like that?
I don't know what you have against keeping x fixed, but you could try dividing the (n+1)th by the nth - which eliminates the x. This gives you $h_{n+1}/h_{n}$ in terms of stuff you know. But you will probably end up needing to know one of the h's.

But if the objective is just to find h, there's no need - it's numbers-in numbers-out.

 

[chrisa88]

I believe your equation is wrong... It should be h_n/d_n = y_n/x giving h_n= (y_n*d_n)/x. Doing it this way, and even your way, would mean that the gap length h_n would get larger for every measurement rather than concluding to one measurement of the gap length h_n... If the gap length does not change for every measurement taken (besides when measurements are taken for trial 2) then every measurement at each d_n should conclude to one measurement h.

*edit: I wrote the first equation wrong.. derp.. but the following equation was correct

 

[Simon Bridge]

Oh yeah - I got it upside down, well spotted :)
Trying to use your notation I keep getting x and y mixed up.[*]

h and y should be in the numerator. You still get y getting smaller as d gets bigger
what about the questions?

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[*] I would normally use D and d for the far and near distances, L would be the length to be found and l the measurement on the ruler. Something like that. So: L/D = l/d