i was reading this thread and to me it seems as if you are making it more complicated than it has to be. maybe i missed some important information, but each reciever has an infinint amount of points that are x seconds away (only infinint in a technical sense. in reality there are just a lot. on a graph, there would only be a few, depending on the distance and the size of the grid. but you know what i mean.) so if it takes x time for the signal to get to the reciever, then all the points that are y distance away (y being how far sound travels in x time) would form a circle. if you were to make the circle for two points, one of 4 things would happen:
a. the two circles are identical. this is not possible since for that to happen they would have to have the same center, and the 4 recievers do not have the same center, so we can eliminate this outcome
b. the two circles do not intersect at all. in this case there would be no answer, and the problem would be invalid. im assuming that the problem has a solution, so we can eliminate this outcome
c. the two circles cross at one point. i believe this can only happen if the sender is in the dead center, in which case the times would be the same for the two recievers. if that were the case, it would be immediatly obvious, and we would have no need to even do any solving, so we can eliminate this as a possibility.
d. the two circles cross at two points. this is only other possibility. so after we find out where they cross, all we need is to make a circle of possibilities at a third reciever, and we have the solution.
so did i miss something and oversimplify, or what? cuz all that stuff about the arc and physics and what not seems like its overkill on this problem. as for a spefic answer, i dont have one yet, but i didnt want to come up with a solution if im not even solving the problem right.
