How to find the signal vector in the plane?

Question

Geometry

There are 3 points A, B, C which form a regular triangle with a side of 8 cm.
Outside this triangle there is another point (D) that emits a signal. The signal propagates in all directions, equidistant from its source (point D) and moves at a speed of 10 cm per second. In a certain unit of time T(a) = 17:28:10:328, the signal reaches point A and it fixes the arrival time, after some time the signal reaches point B, which also saves the detection time to the variable T(b) = 17:28:12:628 and finally the signal reaches point C, the time is T(c) = 17:28:17:428. Question: Is it possible to determine the signal vector relative to the median of the triangle based on these data? If not, can it be done with some kind of angular error? Sector 30 degrees, 45, 60, etc. (of course, the smaller the better).

17:28:12:628 - hours:minutes:seconds:milliseconds.

Answer 1

GPS solves a similar problem (if you forget about all sorts of physical effects).
You will have 3 variables - signal source coordinates (x, y) and signal sending time (t).
You are given the coordinates of three points (x_i, y_i, i=0..2), three signal acquisition times (t_1, t_2, t_3) and the signal speed (v).
Convert the time to seconds relative to the minimum of the three times (after all, you only need relative delays) so that the numbers are not too large. Those. the minimum of the three times will be 0, and the other two will be the difference from this time.
Equations that the signal travels a given distance in a given time with a known speed:
(x_i-x)^2+(y_i-y)^2 = ((t-t_i)*v)^2
You can count the time not in seconds, but in 1 / v, then the equations are slightly simplified (the coefficient before t ^ 2 is everywhere 1, not v ^ 2).
can be solved analytically. Subtract the first equation from the other two. You will get 2 linear equations with three unknowns x, y, t. Consider that t is a constant and solve equations for x and y (Via determinants, or Krammer's method). You will have some kind of linear dependence of x and y on t (large formulas, yes). You can simplify the calculations if you first write the equations in the form A1x+B1y=C1+D1t.
Then substitute these dependencies into the first equation and you will have a quadratic equation for t.
Solve it. Substitute t into the known equations for x and y and there's your center (you also know when the signal was poisoned).
Of the two values ​​of t, one will be in the future (positive), it will have to be discarded.