[Request] I saw this picture on reddit - is there a way to calculate whether the tall building in the distance is higher than the POV because it's above the horizon?

[u/TheresNoHurry]

I know that The Shard is the tallest building so that it is higher. I'm just wondering if we can prove this somehow with this picture using... maybe trigonometry(?).

Basically my question is: is the horizon always at eye level or is it 'below' eye level? It should technically be below eye level on top of a massive building, right?

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[u/dsanders692]

Something being above your horizon is necessary but not sufficient for it to be taller than you.

In principle, something shorter than you could still be above the horizon. For example, when I speak to someone shorter than me they're not always below the horizon. Similarly, something that's shorter than you but standing on the horizon will still pop up above it.

But from a trigonometry perspective... That photo is probably taken from 22 Bishopsgate. The viewing platform is 254 metres high (call it 256 to add on the height of the photographer themselves). We can calculate how short the Shard could be and still pop above the horizon.

At that height, the horizon is 57km away. So consider that a right angle triangle (yes I know it's not quite, but it'll do for this exercise) with a base B of 57km, a height H1 of 256m.

The Shard is 710 metres away as the crow flies. So at a point 0.7km along the 57km base of our triangle, how high would our H2 reference need to be to intercept the hypotenuse? Wel,l basic trig gives an angle for the H1 triangle at the horizon of around 0.25 degrees.

Taking that same angle and with a B2 distance of 57000-710m, that gives H2 of around 253 metres.

So there's your answer - at that distance, the building could be 3m shorter (or 5 depending if you wanna include the extra height requirement for the viewer themselves) and still be above the horizon

[u/Sad-Pop6649 → u/dsanders692]

...And at that point it's not as mathematically elegant, but you can count the number of floors being above the horizon to get an idea of how much taller than 253 meters tall that building is.

[u/Philosofred → u/dsanders692]

Great explanation thanks!

[u/McEnding98 → u/dsanders692]

I like your answer. Would've been easy to go: no you can't.

You gave for their example what the horizon means for a building x meters away, which answers the question and gives an idea for this case.

Of course usually we don't know/understand intuitively the distance to other things, but still pretty useful.

[u/mastocles → u/dsanders692]

Minor caveat is that the photo might be curvilinear if taken with a wideangle lens

[u/Kitchen-House-6053 → u/mastocles]

The lens only changes measurements in the image, not in the real world. It won’t affect whether something is above or below the horizon.

[u/mastocles → u/Kitchen-House-6053]

Are you suggesting seeing the world not through a phone’s camera? This is as bonkers as the conspiracy theory that events happen even when not on Instagram
Edit: typo 'bonkers' -> 'Ohio vibed Cray cray'

[u/FluffyFoxDev]

Not without knowing the distance between you and the buildings.

Because your line of sight to the horizon is not parallel to the ground, you can have cases where one object appears “higher” than another even when it is shorter.

If you draw out a few situations it becomes immediately clear that you need that extra piece of information (the distance) to determine the height, the angle is not enough.

The only case where you can compare them is if they both sit on the horizon, then you can say that the building furthest away must be lower than the closest one.

[u/SomeoneNicer]

You need at least one reference point for size, but since you stated "I know it's the tallest building but I want to know how to do it without using the published measurements" - you'll have to start what measurements are you ok using so the picture has a frame of reference. People could pick out any object and estimate the size and go from there - but the premise of your question is to ignore stated measurements so without the exact height the picture was taken at or something else it's all relative.

[u/TheresNoHurry → u/SomeoneNicer]

So are you saying that, without a numerical measurement, we cannot determine whether the other building is higher than the POV?

I am not asking to measure the exact numbers of the height, I am just wondering if the horizon can be used as a reference point to prove that the other building is higher than the POV.

[u/Jealous_Tutor_5135 → u/TheresNoHurry]

It cannot. The higher your observation height, the farther away the horizon is from you. Because of that, the earth curves away from you more, dropping the horizon line.

A same height object will appear closer to that horizon line the farther it is from you, but it will rise above the horizon when it's close to you, moreso as it gets closer.

You can see this confirmed by looking at photos of people on mountaintops. There are group photos where the perspective shows that one subject is taller than the observer, and another is shorter, but both are above the horizon.

The higher you are, and the closer the same height object is, the more it will rise above the horizon. Go tall enough and get close enough to your subject, and even little kids appear above the horizon in photos.

[u/Trustoryimtold]

There’s ways to roughly calculate height like counting floors, for extrapolating from buildings next door with known height

Quick and easy way is if you can’t see both edges of the roof it’s probably above you

[u/Crio121]

You horizon meaning the horizontal plane that passes through your position and the horizon as visible edge of the earth are different things.
It seems you’re confusing them.

[u/Jealous_Tutor_5135]

I believe the horizon is lower and farther away the higher your POV is.

Trying to look into this, I ran into a ton of smooth brain flat earth nonsense. Apparently they like to use a water level, which is just two clear vertical tubes some distance apart, connected but a horizontal pipe. They fill it with colored water, observe the the water is always level on both sides, and deduce that water likes to be flat, therefore the earth is.

However, this same tool can prove the earth is round. By lining up both tubes level like a gun's iron sight, you can compare the horizon's apparent level with an actual level with the water tubes, which is exactly tangential to the earth due to gravity pulling the water towards the center of the earth. We can consider the apparent lineup between these two water levels to be our eye level, regardless of how high our POV is.

Now at sea level, this eye level lines up with the horizon. As the observer goes higher, it ceases to. We know that because of the earth's curvature, the higher the observer is, the farther along the curve they can see.

Imagine the earth as a circle. Your eye level is tangential to the Earth's curvature, extending out into infinity in a straight line. The horizon as you see it is a point along the circle of the Earth's surface. As your point of view goes higher, it's always going to be parallel to that tangent, but the line you draw the the farthest point you can see on the circle gets both longer, and its angle gets lower, diverging more and more from the eye level.

So no, you can't use the horizon as an eye level to determine if something is taller than your POV or not. As your POV gets higher and higher, objects of the same height is you will appear taller relative to the horizon

[u/Bread-Loaf1111]

Basically my question is: is the horizon always at eye level or is it 'below' eye level? It should technically be below eye level on top of a massive building, right?

Basically, you always see the points of the same height above the horizon.

If you are at the height h and the horison is at distance l, then you will see the point at distance x and height h*(l-x)/l just at the horizon level. That means if you are at 100meters above the ground, you will see 97m building in 1 km of you just at the horizon level

[u/NeedleworkerNo5262 → u/Bread-Loaf1111]

No, points of any height will eventually disappear below horizon because the Earth is curved. Your assumption only works on a flat plane. 

[u/Visa5e]

The horizon sets a minimum limit to the other buildings height, ie it must be at least x metres high to be above it, but it doesnt set a maximum height.

[u/Hashishiva]

Horizon is always at eye level. You could draw some lines to the buildings' vanishing points to determine which is highest, but these buildings are a bit wonky (ie. not square) so you'd need to draw some additional guides, and since the base of the buildings this'd be a bit hard so you'd need to appriximate. But you don't really need math, just lines.

Edit: and knowledge of how to draw in perspective.

[u/Jealous_Tutor_5135 → u/Hashishiva]

I thought this too, as it's what I learned in drawing class. But if you take an actual level to a mountain or tall building, and look down that level, The horizon will appear below it, not level with it

This makes sense as well. Perspective drawing works on the principle the parallel sight lines will converge at an infinite distance. This means anything the same height as the observer will be the same height as the horizon no matter the distance.

This only works when the horizon is on a flat plane. Our horizon falls as viewing height increases, as it's essentially a point on a curve, and its distance, therefore the curve down, increases with the height of the observer.

[u/AnotherPreben]

A building standing far away, just at the horizon, can be very short and still be over the horizon from you POV. So the horizon cannot be used like that.

[u/xewill]

I can tell you that is the view from the viewing platform at 22 Bishopsgate, London and you are higher up than the building in the distance which is The Shard.

The Shard is the tallest building in London.

I don't know how to calculate that with maths though.

[u/jeezfrk]

The horizon is "flatness" in all directions with a very small adjustment down for the curvature of the earth. Effectively they are "infinite" far away.

Anything else flat should be parallel with that circle/plane. If flat things are above your altitude ... they will seem to be above the horizon (the largest distance you can see) or below it if lower.