A shot is fired from the ground to a helicopter in flight.  Using three microphones mounted on the helicopter, determine the location of the shooter.

Compliments of NASA

Compliments of

Compliments of

Add a fourth microphone on the helicopter to signify which side of the plane the gunfire originates. Also, adding the helicopter's altitude can help back-triangulate the source of the gunshot.


The bullet rotates once between the first and third microphone, and twenty times on the trip to the second microphone.  This is why t1 and t3 are so close together.

To calculate t2 we need only to be concerned with the u component along the k axis and not the rotational behavior.  The rotational properties show their colors on the trip to the second microphone with twenty rotations.

Thus, the process in this example deduces the backward problem and its solution.  In practice only time differencest12, t13 and t23 are harvested from the electronics and the solution must be solved using the cubic and the quartic, creating lengthy but not impossible equations.  The procedure is to develop the f(x) equation of one variable in x, graph the curve and determine where it crosses the x-axis.  Once this is found, back substituting precipitates the solution for u, the unit vector of the direction of the line of fire.  Then the position vector of any point on the conical spiral can be determined and advanced to find the coordinates of the shooter.  A fourth microphone is placed to determine which side of the plane is the origin of the projectile and using trigonometry with the helicopter's altitude, the souce of the projectile can be calculated.

The rifling in the barrel causes the bullet to spin for more accuracy.  The solution won't work for a shotgun.

In memory of Professor Szoke of the University of Akron and his WWII stories of the bombings driving his nose to the grindstone immersed in mathematics for solace.  He imparted an inspiration and enthusiasm to me that I likewise have engrossed my attention in mathematics during troubled times in my past.