← THE SHADOW WATCHAnnex 01 / inverse optics / coded flashes

Field correction / after the third watch

Put another lamp
on the tower.

A shadow can appear before a bird passes the lighthouse without arriving from the future. The bird might already be out in the fog. The ground may be showing us its unseen present.

Two-colour shadow clock

Simulated measurements · level tidal flat

Set the unseen bird in motion

Lamp A: 40 m high. Lamp B: 30 m high. They flash in coded succession at each instant, two seconds apart.

Two-lamp lighthouse shadow geometryA 40-metre and a 30-metre lamp cast separate coded shadows of a bird onto level sand. The rays intersect at the bird's present position. 0200 m400600800 A / 40 mB / 30 m A B INFERRED BIRD LOOK AT THE SAND.
Frame 1 / t = 0 sTwo shadows, one bird · pan on narrow screens ↔
Shadow A, now304 mFrom 40 m lamp
Shadow B, now456 mFrom 30 m lamp
Reconstructed bird—Back-trace both rays
Arrival at tower, from t=0—Constant horizontal speed

The inverse problem

The ground has coordinates.

Let the tower stand at x = 0, and let a bird at range x and altitude z interrupt a lamp at height H. Similar triangles put its shadow farther down the tidal flat:

S(H) = Hx / (H − z)z = (Sₐ − Sᵦ) / (Sₐ/Hₐ − Sᵦ/Hᵦ)x = Sₐ (1 − z/Hₐ)

At fixed altitude, shadow distance and shadow speed are stretched by the same factor; their ratio gives time-to-tower. But a climbing bird breaks that one-lamp clock. Two different lamp heights let the rays intersect and recover x and z at each flash.

Default: A goes 304 → 272 m and B goes 456 → 408 m in two seconds. Both clocks say 19 seconds.

Precision costs something: with independent Gaussian shadow-position noise of σ = 1 m, 20,000 seeded simulations put the two-lamp arrival estimate mostly between 16.2 and 23.0 seconds (central 95%). The bird is not a stopwatch.

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