Every camera sees the ball only as a 2D pixel โ a single ray in 3D space, not a point. Automated line-calling systems (the mechanism behind Hawk-Eye-style goal-line and tennis line technology) reconstruct the true 3D bounce point by intersecting several such rays, one per synchronised camera, at the instant the ball touches the ground.
With perfect cameras the rays would meet exactly. In reality every pixel has finite size, so each ray carries a small angular uncertainty and the rays are skew โ they miss each other. The system instead finds the point p that minimises the total squared perpendicular distance to all N rays:
ray i: points o_i + tยทd_i (d_i unit direction)
minimise ฮฃ_i |(I โ d_i d_iแต)(p โ o_i)|ยฒ
โ solve [ ฮฃ_i (I โ d_i d_iแต) ] p = ฮฃ_i (I โ d_i d_iแต) o_i
a linear 3ร3 system built purely from the ray geometry.
- Cameras โ more independent viewpoints over-constrain the system and average out pixel noise; real systems use 6โ10 synchronised cameras at 300+ fps.
- Pixel noise โ each ray's direction is perturbed by a small random angle before triangulation, modelling lens/sensor pixel quantisation.
- Target offset โ where the ball actually bounces relative to the line (negative = clearly in, positive = clearly out, near 0 = the contested case).
- A ball of radius r counts as IN if any part of it touches the in-bounds side, i.e. (bounce offset โ r) โค 0 โ the same rule real line judges use.
- Ray residual (RMS) โ how tightly the rays actually converge on the solved point; a large residual is the system's own confidence signal that a call might be wrong.
Run a trial near 0 mm with camera count low or noise high and watch the system call flip away from the ground truth โ exactly the failure mode camera count and synchronisation are engineered to avoid.