# The horizon, concave vs convex

How would a horizon inside of a ring or a toroid look like?

As if a planet's surface was concave, instead of convex, how far away would the horizon appear? Im trying to get a handle on what the inside of a ring, or a toroid would look like.

• Reminder to Close Voters: The OP can't fix problems if he is not aware of them. Please give some tips on how to improve the question if you are voting to temporarily put it on hold. @Bert: Someone voted to put this question on hold as "Too Broad". If 5 community members vote as such it gets put on hold until you edit it - which brings it to a reopen queue needing 5 other people. I don't know much about the topic: how big is your ring (radius)? How do you imagine the ring? What do you think will happen? – Sec SE - clear Monica's name Jan 3 '18 at 12:08
• Questions on this site need to be specific and answerable, You're asking about a toroid planet, and a hollow planet. The answer is going to be very different depending on which type of planet we're talking about. Questions on this site need to be specific. It would also help to specify the size of the planet you are asking about. – sphennings Jan 3 '18 at 12:48
• @sphennings I was able to provide a simple answer that covers all cases. The size of the planet is just a variable in a short formula, so I don't think it is required. – Renan Jan 3 '18 at 13:29
• Ever played Halo? – bendl Jan 3 '18 at 14:55
• I suggest that you download a 3D modeling program (like Blender), generate a torus (this can be done with a builtin tool), and move the camera around to obtain various perspectives. – Display Name Jan 3 '18 at 21:54

## 2 Answers

How far the horizon appears does not depend solely on the shape of the planet you are on. It also depends on its size. If you are also considering shapes that are not spherical nor spheroids, then the actual shape also matters.

For a ring world, there is a question about the visibility among inner points of the ring. It also applies to a hollow world:

Visibility in a ringworld atmosphere?

For a toroidal world, the distance to the horizon would be pretty much like that on Earth... Just consider the radius of the cross-section of the ring to be like the radius of a planet, and then calculate accordingly. The usual formula is, according to Wikipedia:

$$d = (2Rh + h^2)^{\frac{1}{2}}$$

Where d is the distance to the horizon, R is the planet (or torus cross-section) radius and h is the height of the observer above the mean altitude, which happens to be sea level on Earth. For other planets, specially dry ones, consider this (taken from the link):

On other planets that lack a liquid ocean, planetologists can calculate a "mean altitude" by averaging the heights of all points on the surface. This altitude, sometimes referred to as a "sea level", serves equivalently as a reference for the height of planetary features.

• I think this goes into good detail if you're on the outside of the ring, but unless I'm mistaken there would be no horizon at all if you're on the inside, which is almost definitely where you're going to be – bendl Jan 3 '18 at 17:59
• @bendl if the planet is a concave ring, yes, it's just like being inside a crater. Otherwise, for a perfect or near perfect Torus, it is possible ot have an horizon. – Renan Jan 3 '18 at 18:11
• Oh I understand your point now; the horizon would be normal to the torus, not tangential. – bendl Jan 3 '18 at 18:14

I don't horizon is applicable. If you are on the inside (concave surface) then either you can see the entire interior for small structures, or your visual distances limited by the transparency of the atmosphere.

Niven's Ringworld had a thin atmosphere, which allowed you to see the rest of the ring once you were far enough away. The illustration isn't right, as you wouldn't be able to see the start of the curve.

Locally you would see just the surface vanishing into the haze, then "The Arch of Heaven going from one side of the sky to the other.

Ringworld had thousand mile high mountains on either edge. Once you were far enough from the rim that the slant path through the bottom half of the atmosphere (18,000 feet on earth) was more than about 50 miles, the mountains would be invisible. Call it somewhere between 100 and 300 miles depending on the transparency.

This does NOT address a torus. This is the inside of a ring.

If you were on the inside of a torus you have a different set of circumstances:

• If the torus is spun for gravity then there is a ceiling horizon. This will block vision of the floor further ahead. Note that in this illustration, you can see some of the distant view through the myriad windows on the 'ceiling'

If it's not spun, then the you can stand on the inner surface of the 'donut hole' In this case you would have a conventional horizon dipping down in two directions, and side walls on the other two directions.

• It's definitely worth noting that this is slightly different for a torus. I misunderstood Renan's answer at first but in the case of a torus world there will be a horizon to your left and right in the picture you have above. I'm imagining it would look more like you're standing at the top of a large hill. – bendl Jan 4 '18 at 13:18
• @bendl when I answered this I was imagining a natura Torus, so an observer would be on the outer side of the torus surface. Sherwood's answer reimagines it from the point of view of someone inside the surface of the Torus. +1 for that. – Renan Jan 4 '18 at 20:21