Beacon Chain Relay

What is modelled

The Earth is a sphere, so the ground rises between two distant towers and blocks a straight line of sight. A fire on top of a tower of height h can be seen along the ground out to its horizon; two equal towers can see each other when they stand no farther apart than twice that distance, at which point the line between the fires just touches the ground halfway. Air bends light slightly downwards, so the fire is seen a little farther: the same geometry works with a larger effective Earth radius.

Equations

• Effective radius Re = R / (1 − k), with k about 0.13 for ordinary air.
• Horizon arc d = Re arccos[Re / (Re + h)] ≈ √(2 Re h).
• Maximum spacing of equal towers smax = 2 d; ground rise midway s2 / 8Re.
• Towers n = ⌈D / s⌉ + 1, message time (n − 1) × delay.

Checks run in the page

• Bisection on the sight line to a curved ground against the closed-form horizon.
• Bisection on the grazing line between two towers against twice the horizon.
• A full chain placed tower by tower, every link tested.
• A refracted ray traced over the true radius against the effective-radius formula.

Honest simplifications

The ground is a smooth sphere with no hills, trees or haze, the air bends light by one constant coefficient, the fire is a point and the delay per tower is just a number you choose. Real chains were laid out on high ground and needed clear weather.

• Press Grazing spacing to see the line of sight just touch the ground.
• Press No refraction and watch the maximum spacing shrink.
• Press Long coast to see how many towers a long route needs.