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<   No. 5034   2023-08-17   >

Comic #5034

1 Lambert: Yeah, I can’t see any city on the horizon. Draak must be mistaken.
2 Kyros: You realise the horizon for him is an extra mile or two away because of your relative heights?
3 Mordekai: Surely that would only be the case if, ha ha, the world was round.
4 Kyros: Is Wisdom or Intelligence the dump stat for lovable rogues?
4 Mordekai: Both!

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Some simple geometry allows us to calculate that the distance d to the horizon on a spherical planet is given by the expression:

d = √(2Rh + h2)

where R is the radius of the planet and h is the height of the observer's eyes from the ground. If we assume this world is the same size as Earth, and Lambert's eyes are a metre from the ground, then his horizon will be 3.57 km away, while if Draak's eyes are 2 metres from the ground then his horizon will be 5.05 km away.

Interestingly, this means Lambert can see 40 square kilometres of land, while Draak can see exactly twice as much at 80 square kilometres. So the area of land you can see scales almost exactly as your height from the ground (for small heights relative to the radius of the planet).

This is strictly talking about the horizon of a perfectly spherical planet and ignores: (1) local topography; (2) atmospheric refraction; (3) the fact that a city will likely have some buildings that rise above the horizon, making the tops of its buildings visible from further away.

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