Capture: Got the blues
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5km seems a pretty sort distance for you to be seeing the earth curve.
Truth is stranger than fiction. I looked this one up before I wrote it, but I did also remember that it's a lot closer than one might think, when one is down low  for instance sitting in a lifeboat. My estimation of the height of the camera, which was pressed to my eye, is that it was about 1.8m above sea level (I'm around 1.95m tall). So the horizon is about 5km away. I would be completely unable to see the bottom 1.8m of any object 10km away.
It is very hard to be sure of how far away the boats were, as the height of their masts is unknown. But at full zoom, a similar appearing vessel fills the entire shot at a distance of around 500m. The closest vessel is 422 pixels high in the shot above, out of 3216 pix. So that makes it around 3.7km away. The one beside it comes up only 200 pixels, so it's over twice as far away, if it's mast is roughly the same height. That puts it around 8ish km away, beyond the horizon. This is sort of confirmed by the other shot which is from somewhere between 4 and 5m above sea level, in which the furthest boat seems to be on the horizon. This suggests it's about 78km away at the time of that shot, which is 45mins before the sea level shot. The ship was making good headway out to sea the whole time, so could easily be a further 34 km out by then.
ETA: Which is not to say there isn't mirage effect there. I'm sure there is, because it can be seen in this shot, containing Little Barrier Island, which is something like 5080km beyond. You can see the corner is floating. But notice that this floating is also above the height of the decks of the boat on the horizon, but that is reasonably clear and undistorted. This suggests the mirage effect is happening mostly beyond that point.

BenWilson, in reply to
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Sorry Ben, I don't mean to be a pain about this, but I'm really fascinated.
Not at all, I'm interested too. Feel free to take this offline (we've got each other's emails), unless anyone else is interested to hear about this.

Lilith __, in reply to
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Which is not to say there isn’t mirage effect there. I’m sure there is, because it can be seen in this shot, containing Little Barrier Island, which is something like 5080km beyond. You can see the corner is floating. But notice that this floating is also above the height of the decks of the boat on the horizon, but that is reasonably clear and undistorted. This suggests the mirage effect is happening mostly beyond that point.
Mirages can be horizontally discontinuous, depending on the local temperature gradients in the air between objects and viewer. This is a very dramatic example. Obviously your picture is all sea, but that needn’t mean that the air is the same temperature all over. Or, as you say, it could be different further towards the Island.

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Lilith __, in reply to
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I’m interested too. Feel free to take this offline (we’ve got each other’s emails), unless anyone else is interested to hear about this.
I think I'm all out of wild speculations, actually. ;)
Could that yacht really be beyond the horizon? <boggle> 
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If you are just above sea level eg standing in the water or sitting in a kayak, the horizon does not appear to be as far away as when you are higher up eg on a hill. Isn't that logical, as you are closer to the surface of the earth? You have to be in sea without waves to notice it best.
Anyhow here is another blue vista with Mt Maunganui. 
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In reply to Ben 
Truth is stranger than fiction. I looked this one up before I wrote it,
And what planet are you on? Apparently planet Kulthea! Strange to say  you looked up fiction in the quest for truth! ;)
Fortunately, Kulthea has only a slightly larger radius than planet Earth. So the horizon distances on Earth are near enough those on Kulthea, a little shorter e.g. 4.8 km for your 1.8 m eye height rather than 5.0 km.And David's formula 
I'll back up Ben's general math about horizon distance, the formula is
SquareRoot(eye height above water in cm / 6.752) = distance to horizon in kmcomes from another slightly larger planet than ours too. Replace 6.752 with 7.848 for Earth.
The mathematically inclined are less likely to mistake their planet if working nearer to first principles. Horizon distance y = SQRT (2r*h) where 2r is Earth's diameter (mean diameter 12742 km) and observer/eye height h in the same units  this is the quadratic formula y squared = 2rh as an approximation to the arc of a circle, accurate for practical purposes for all human to mountainscale heights.
Square root of 12742 km * 0.0018 km = 4.79 km.
For David's formulation  100,000 cm/km / 12742 km = 7.848 rather than 6.752And of course  yachts, ships "hulldown" on/over the horizon are absolutely normal everyday stuff, nothing surprising, whereas substantial mirages rely on strongly stratified air temperature variations close to the sea surface, which is not going to happen in the Hauraki Gulf in the middle of a mild onshore breezy day in late summer with warm sea surface temperatures.

Lilith __, in reply to
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And of course – yachts, ships “hulldown” on/over the horizon are absolutely normal everyday stuff, nothing surprising, whereas substantial mirages rely on strongly stratified air temperature variations close to the sea surface, which is not going to happen in the Hauraki Gulf in the middle of a mild onshore breezy day in late summer with warm sea surface temperatures.
I'm not a sailor, so I bow to your superior knowledge of what's common, but a "hulldown" ship is still a mirage. My point was not that this was a sensational happening, but that the apparent horizon was confused by mirages, and might not be where it appeared to be.

BenWilson, in reply to
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And what planet are you on? Apparently planet Kulthea! Strange to say  you looked up fiction in the quest for truth! ;)
Classic! Actually I looked up Yahoo answers first, and only gave that link because of a convenient tabular form, which happened to agree with other answers that I'd checked to within the very loose distances I was using.
accurate for practical purposes for all human to mountainscale heights.
I wasn't keen to model the maths. I've only got so much of that in me per day, and today's is all gone on a physics assignment.

Speaking of persons on another planet, curvature and all that – I was surprised this afternoon to see these curved posters neatly interpolated with those advertising exhibitions on at Tairawhiti Museum.
Coastal peoples were aware of the curvature of the earth and hence its mysterious spherical shape from the time of coastal seafaring, where canoes, small boats and ships gradually disappear over the horizon yet appear again if one climbs a modest sand dune or small hill. The idea that Europeans at least thought Earth to be flat in mediaeval times preColumbus is a lie or a myth if you prefer, invented in the 19th century.
Totally unhelpful to denigrate Climate Change/AGW denialists as flatearthers etc – but you do have to wonder what planet they are on and wonder at the imagery they use on these posters. And why is this *orator*, a stranger to the truth as well as science, given space to advertise a commercial venture on our ratepayerfunded museum/art gallery’s site? I’ll be making enquiries.

BenWilson, in reply to
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I wasn't keen to model the maths. I've only got so much of that in me per day, and today's is all gone on a physics assignment.
OK, that's not true. To help me sleep, I worked this one out to follow the other two derivations here. If you imagine 3 points, your eyes, the center of the earth, and the horizon, then they form a right angled triangle with the right angle on the horizon. The long side is the one from your eyes to the center of the earth, length r + h, where r is the radius of the earth and h is the height of your eyes. One other short side is length r (to the horizon from the center). We want to find the other side, call it c, from your eyes to the horizon. After Pythagoras and some algebra, we get c = sqrt(h^2 + 2rh). r is approximately 40000/(pi*2), just from the original definition of a meter as one ten thousandth of the distance from the North Pole to the Equator (through Paris, I think). I get 6366km. So if h = 0.0018km, then c = 4.78km. This derivation has the advantage of not requiring terrestrial heights. At 100,000km from the Earth, the horizons are 106,175km away. Now I can sleep.

ChrisW, in reply to
Trust you've had a wellearned sleep after that, Ben. I work differently  have to let someone be wrong on the internet after 10pm, and generally (really should!) even if it's myself or my own unfinished business if I want to sleep.
When beyond "terrestrial" heights, it becomes clearer your formulation measuring the eyehorizon straightline distance is not what we usually think of as the distance of interest, the map/chartdistance over the surface of the earth. In the case of your 100,000 km eyeheight, the 'real' horizon distance would be just a little under your quarter of 40,000 km approximation to the earth circumference.
(Metre intended to be a 10 *millionth* of that poleequator meridian  but one of the guys Mechain who did the precision survey work in the 1790s lost much sleep over an error he knew about  went back to northern Spain to fix it and died in the attempt. Good story, read the book  lost in the mental archives for now. Upshot  the standard metre was a poorer approximation to the earth's dimension than intended, 6366 km not the best radius to use. But good for checking the numbers from first principles!)
Not even blueskies research!


JacksonP, in reply to
here is another blue vista with Mt Maunganui.
That's a great shot Hilary. Thanks.


JacksonP, in reply to
Lake Paringa blues again – sky, shadows and reflections thereof.
Much betterer. Great shots. ;)

ChrisW, in reply to
Pure blueskies research.

BenWilson, in reply to
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