For each detected air shower, the Fly's Eye records the pattern of fired PMTs. Each hit PMT also stores the time of arrival of the light pulse from the shower, and the integrated amount of light received. From the pointing directions and the arrival times of the PMT hits, it is possible to reconstruct the trajectory of the air shower. The procedure is traditionally divided into two steps:
For the "plane fit", we treat the air shower trajectory as a straight line. Strictly speaking, the shower has a lateral size of the order of ~100 m, depending on the stage of shower development. At sufficiently large distances (more than a few km), the light from the shower would appear to come from a line source. We also treat the detector itself as a single point, even though it is spread over a ~100 m wide area. Again, this size is negligible at distances of a few km or more. The point of the detector and line of the shower together defines a plane which is referred to as the "detector-shower plane", as illustrated in the figure.
The parameters of the shower-detector plane can be determined by fitting the direction cosines of the hit pixels to a plane. It is customary in this procedure to choose the detector site as the origin and fit for the normal unit vector of the shower-detector plane.
Next, the exact trajectory of the shower within the shower-detector plane can be specified by two additional parameters: (a) the impact parameter Rp, and (b) the incline angle c0, or its complementary angle y. It turns out that there exists a simple trigonometric relationship between the arrival times ti and the angles ci of the hit pixels. The angle ci for a given tube is measured from the position where the shower strikes the ground (called the "core" location) to the point of observation P. This relationship is given in the diagram below. This analytic formula involves Rp and c0 as parameters, plus an arbitrary time offset t0, and relies on the approximation that both the air shower particles and the light emitted travel at the speed of light in vacuum, c.
The parameters Rp, c0, and t0 are determined from the fit to the above formula. This step is usually referred to as the "timing fit".