User blog comment:Shadow at Morning/Determination of the Size of Vale via Cartographic Analysis/@comment-226878-20150818022744/@comment-24891101-20150818044343

There's actually an error here that I didn't notice. 1800km is not the distance to the horzion, but the arclength (distance along the ground) of that sector. You actually have to be 295km up to see a horizon of arclength 1800 km away. That's to the horizon. With an object of any height, you simply add their horizon distances to see if they can see each other, because the horizon line is tangent to the planet and so they can just barely see each other.

It still doesn't work. Even if Signal is ten times the height of the Burj Khalifa, they'd stil have to be 200 km up to see it.

And none of this takes into account visibility conditions or the limits of the human eye.

-

Distance to horizon: d=sqrt(2Rh+h²) [R is radius of planet, h is height of vantage point]

Conversion of d to arclength: l=R*arcsin(d/(R+h))

So we have: 1800=l=R*arcsin(d_s/(R+h_s))+R*arcsin(d_r/(R+h_r)) [d_s is distnace to horizon from Signal, h_s is height of Signal, d_r is distance to horizon from Ruby's position, h_r is altitude of Ruby]