Focus Rescue

Every frame the game turns the optic position into an object distance dₒ, solves dᵢ = f·dₒ/(dₒ − f), and draws the image exactly there. If you can predict dᵢ on paper you can place the optic first time.

• Where it lands. 1/f = 1/dₒ + 1/dᵢ. Push dₒ down toward f and the denominator collapses, so dᵢ sprints off to infinity.
• How big, which way up. m = −dᵢ/dₒ = hᵢ/hₒ. A real image (dᵢ > 0) always has negative m, so it hangs upside down.
• Two answers, not one. Fix the beacon-to-plate distance L = dₒ + dᵢ and the equation becomes dₒ² − Ldₒ + fL = 0. Its two roots are the two lens positions that both give a sharp image, and their magnifications are exact reciprocals — that is levels 2 and 3. Real roots need L ≥ 4f, which is why one lens in level 1 simply cannot reach.
• Virtual and upright. Bring the beacon inside f and dᵢ goes negative: the image jumps to the object side, turns upright, and cannot be caught on a plate. That is a magnifying glass, and level 4 asks for one at the relaxed eye’s near point of 25 cm.
• Mirrors use the same equation with f = R/2, except dₒ and dᵢ are measured on the same side, so the image comes back toward the beacon. Two lenses in a tube extend the same algebra to a telescope, where the angular magnification is M = f_o/f_e with the tube length f_o + f_e.
• Why fine tuning is hard on some settings. Differentiating gives d(dᵢ)/d(dₒ) = −m², so near a magnifying conjugate the image moves m² times faster than the optic. A shrinking setting is calm; an enlarging one is twitchy.

Conventions: distances in centimetres, dₒ > 0 in front of the optic, dᵢ > 0 for a real image, f > 0 for a converging lens or concave mirror. The clear aperture is taken as 3.0 cm, which is what sets the size of the blur on the plate — a defocus of Δ spreads the light over about A·Δ/|dᵢ|. Focus lock reads 100% when the image plane sits on the target and 60% at the edge of the level’s tolerance, and magnification must land within 8%. Thin-lens, paraxial: no aberration, no thickness, no dispersion.