[Sharlin]

It’s mostly implicit. See how the spheres are symmetric about the centerpoint? (Well, almost, the camera is at Y = -0.1.) The (X, Y) coordinates of their centers are simply (-1, -1) and (1, 1), given by the `I = SGN U` variable at the end of line 40. The Z is implicitly zero. Their radius is 1. The plane is basically defined by `P = Y - 2` at line 60.

[Sharlin]

…to elucidate a bit, when tracing the rays on the left side of the image, the sign of u is -1, so we're trying to intersect a sphere at (x-1, y-1, z), and on the right side similarly one at (x+1, y+1, z). This works because none of the left-side rays can even in theory hit the right-side sphere and vice versa. It's a very simple version of space partitioning optimization. And when tracing the reflected rays, we flip the sign of i, because there's no self-reflection – the left-hand sphere can only reflect the right-hand one and vice versa.

[kragen]

these comments were extremely helpful, thank you

[Sharlin]

No problem, this was a fun puzzle to solve =D