This should be an easy question for you guys - triangulation?

[almo]

Hi.

I found a related thread on this topic, but it didnt seem like anyone
actually gave the answer, but I'll pose the question in the same
manner. 2d plane, x,y coordinates, like a football field. 120 foot
diameter circle. 3 acoustic sensors on the circle's perimeter, equally
spaced at 120 degrees apart. Someone fires a gun inside that
perimeter. Assume the speed of sound on that day is 761 mph. 761 mph
* 5280 ft/mile / 3600 sec/hour = 1116 ft/sec. I can determine the
exact instant that each sensor detected the gunshot. How do I
calculate the position of the gun that was fired in x,y coordinates in
feet? Or, to save one step, just the polar coordinates. (Is this a
quadratic??)

TIA,
Alan
almoREMOV...@yahoo.com

[Jeremy Boden]

In message <1150590999.5...@c74g2000cwc.googlegroups.com>, almo
<almo...@yahoo.com> writes

Assuming it is a calm day...
Any two sensors could be used (draw a triangle) to give a result with
two possible answers; the third sensor would determine which result is
correct.

BTW you don't need to restrict your artillery practice to within the
perimeter, unless this is for safety reasons.

--
Jeremy Boden

[matt....@gmail.com]

The simpliest way that I figure you could solve this problem would be
this. Examine each sensor individually. Using the known velocity of the
sound and the time it takes to reach each sensor, you can calculate the
distance from each sensor's linear distance to the gun. Now, trace a
circle around each sensor, with each corresponding linear distance to
the gun serving as the radius of the circle. Doing this, you should see
the point where all three circles intersect. This is the location that
the gun was fired. I hope this helps you. Take care.

Matt

[John Meglier]

<matt....@gmail.com> wrote in message
news:1150593198.5...@y41g2000cwy.googlegroups.com...

each pair of sensors will have a set of hyperbolic lines that are solutions,
the third sensor is needed to resolve an ambiguity
The rest is easy.
Also you do not know when the gun was initially fired, only when the first
sensor detects it.
Police have gun sensors in many cities, they have 6 mics on them, looks like
a big cross, and are located about 2 miles apart, and they can locate the
gunshot region.

[almo]

Well, I was going to use 4 sensors, but my friend's a math wiz, except
not when he's on his way out the door from work on a friday afternoon,
but he told me to use 3 sensors because 4 would be over-deterministic,
and with 3 sensors I'd get a quadratic, but he couldn't give me the
equation off the top of his head. I can even determine the time that
the sound originates, since I control that, (I'm doing a real-world
application) but a more elegant (cheaper) approach would be the 3
sensors. I could figure it out myself, but I'm not into reinventing
the wheel, because I don't get paid for reinventing. I'll talk to my
buddy today to see exactly what he had in mind, although what I've read
here is enough to go on. BTW, if my friend gives me a nifty equation,
I'll post it.

Thanks.

[Dave L. Renfro]

almo wrote (in part):

> 3 acoustic sensors on the circle's perimeter, equally spaced
> at 120 degrees apart. Someone fires a gun inside that perimeter.
> Assume the speed of sound on that day is 761 mph.
> 761 mph * 5280 ft/mile / 3600 sec/hour = 1116 ft/sec.
> I can determine the exact instant that each sensor detected
> the gunshot. How do I calculate the position of the gun that
> was fired in x,y coordinates in feet?

A nice article on this topic is:

Harris F. Mac Neish, "Two methods of locating the
German super gun", School Science and Mathematics 18 #7
(October 1918), 626-628.

He solves this by using geometrical considerations based
on circles and geometrical considerations based on hyperbolas.
I posted the complete text of this short article at:

http://mathforum.org/kb/message.jspa?messageID=4685296

Another useful reference is:

Haroutune Mugurditch Dadourian, "Acoustic circles",
American Mathematical Monthly 28 #3 (March 1921),
111-114. [JFM 48.0678.05]
http://www.emis.de/cgi-bin/JFM-item?48.0678.05
(The JFM data-base incorrectly lists this as Volume 29.)

Dave L. Renfro

[Dave L. Renfro]

almo wrote (in part):

>> 3 acoustic sensors on the circle's perimeter, equally spaced
>> at 120 degrees apart. Someone fires a gun inside that perimeter.
>> Assume the speed of sound on that day is 761 mph.
>> 761 mph * 5280 ft/mile / 3600 sec/hour = 1116 ft/sec.
>> I can determine the exact instant that each sensor detected
>> the gunshot. How do I calculate the position of the gun that
>> was fired in x,y coordinates in feet?

Dave L. Renfro wrote:

> A nice article on this topic is:
>
> Harris F. Mac Neish, "Two methods of locating the
> German super gun", School Science and Mathematics 18 #7
> (October 1918), 626-628.
>
> He solves this by using geometrical considerations based
> on circles and geometrical considerations based on hyperbolas.
> I posted the complete text of this short article at:
>
> http://mathforum.org/kb/message.jspa?messageID=4685296
>
> Another useful reference is:
>
> Haroutune Mugurditch Dadourian, "Acoustic circles",
> American Mathematical Monthly 28 #3 (March 1921),
> 111-114. [JFM 48.0678.05]
> http://www.emis.de/cgi-bin/JFM-item?48.0678.05
> (The JFM data-base incorrectly lists this as Volume 29.)

Another addition to this list of references:

Norman Anning, "Solution to Monthly Problem 2873"
[proposed by D. H. Richert], American Mathematical
Monthly 33 #5 (May 1926), 282-283. [After Anning's
solution there is a note by Otto Dunkel.]

Statement of Problem: At B is the enemy's battery.
At M_1 a battery is to be placed to silence B.
Listening posts are installed at M_1, M_2, M_3,
all provided with stop-watches. From the maps
at hand, the three sides of the triangle M_1M_2M_3
are known. B is not visible from any one of the
points M_1, M_2, M_3. The sound of a gun fired
at B reaches M_1 at the time T, and M_2 at the
time T + r_1 sec., and it reaches M_3 at the
time T + r_2 sec. How far is B from M_1?

Dave L. Renfro