The hyperbola is the inverse function of the quarter circle, a fact of geometry.
The rod is only a visual aid to indicate a rigid device moving with A, and represents the photon path as imagined by A. It helps with those new to SR. The photon doesn't hit the rod. The successful photon has to have the correct angle to intercept the A mirror. How would it 'know' this? It doesn't. As stated, the successful photon has a horizontal vt component that equals the speed of the clock. The vertical st component moves toward the mirror but at less than c. The intersection of the arc of photons and the path/(rod) to the mirror is determined by the speed of A. The angle of the ct photon relative to vertical is a function of v/c. A simple examination shows, a lesser speed, a smaller angle and a greater s,
a greater speed, a greater angle and a lesser s.
The drawing is intended to show why 1 tick for the A clock is longer than 1 tick for the U clock for the general case, which it does. The reason is the constant and independent speed of light. The degree of dilation is a function of v/c. If the path is extended, it will not show anything new. Gamma is still the ratio c/s or (A time)/(U time).