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I'm working on a project and I'm a developer. the math is a bit, well, way beyond me. I can visualize things enough to see that they should work, but that's as far as my brain can take me on this one.

For this project, there are three points in a triangle. Each point has an origin for $xyz$, and an angle that'll make a cone. the direction will be towards one of the points in the triangle (based on some outside factors which one). Based on the project, these three cones should have a single intercept point between all three cones.

Here's a visualization of my goal

I've illustrated this by showing the red dots, which are the origins. The green dot would be the intercept point. My math skills aren't worldly enough to know where to look on how to solve something like this, so I'm hoping someone here and point me in the right direction!

For this, the size of the cone likely won't be more than $500$ ft, but for this application would stop at the intersection point and should be variable.

As for the intersection point itself, I'm trying to find a boundary intersection.

A.D
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John Sly
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  • When you say "based on the project, these three cones should have a single intercept point between all three cones", what does this mean? That when you're given the points, directions, and cone angles, there's guaranteed to be a single solution? Or that you have to adjust the cone angles so that this is true? or what? In general, this sort of problem's likely to be really tough: even if there is a unique solution, finding it may involve solving a 6th degree equation. I encourage you to write out very clearly what's given and what's wanted. Then maybe we can help out. – John Hughes Jun 25 '15 at 15:56
  • Also: do you need an exact solution, or one that's numerically correct within some modest tolerance? The problem looks readily amenable to some sort of numerical method, but the answer you get back from this will only be correct to some (pre-chosen) accuracy, i.e., you can say "find me a point that's withing 1/100 of an inch of the exact intersection". – John Hughes Jun 25 '15 at 15:58
  • Those are good points. This projects is finding the source of something right now and there are a few kinks I need to work out still on the practical application. It's more of a "lets see if we can do this". As for the direction of the cones... each point on the triangle is a "listening point". based on the difference in times they hear what they're looking for, they generate an angle. If they hear it at the same time, then the cone would be a perfect disc. since that's unlikely to happen exactly, the angle will move towards the listening point that heard it first. Since there's a source... – John Sly Jun 25 '15 at 16:04
  • continued: Since there's a source all three generated cones should have an intersect point. These won't account perfectly for determining the Z position, since all three listeners are on a flat plane, but for the application of this, it won't matter (and once it's tested to work, it can be added to based on the initial concept). In a perfect world, there should be an absolute answer, but since this will come down to the accuracy of my measuring equipment, if there's a way to find it within a tolerance, that'd be ideal. As far as asking more specifically than that, I'm not sure I know how to. – John Sly Jun 25 '15 at 16:07
  • If you're trying to locate the source of a sound/radio wave/whatever that propagates with constant speed, and you know the time differences of the arrival of the signal at three points, then you should be looking for intersections of hyperbolae; a quick study of how LORAN navigation works might be a good first step. – John Hughes Jun 25 '15 at 16:08
  • From what I'm seeing is, LORAN systems are on a 2d plane. In this instance. While I can't say to what degree, that'd definitely reduce the accuracy of what I'm looking for. But this does give me some more direction. – John Sly Jun 25 '15 at 16:30

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