Fun With Science  /  Globe Deconstruction  /  Q4 · page 47

The Mirror Settles Nothing. The Same Photograph Does.

Answering Q4 (p.47) and the reflection chapter (pp.113–116) of Globe Deconstruction?

Do not measure the reflection’s shape. Measure the angle from the summit down to the reflected summit.

Levi Miller asks whether perfectly mirrored reflections survive over large water as observer height increases, and proposes to find out with a drone over Lake Pukaki, Aoraki/Mount Cook behind it. The answer is yes — about four per cent at a drone’s ceiling, a thing you measure rather than see — while a single frame from the shore carries a signal seven times bigger in the angle from the mountain to its own reflection: thirty-five arcminutes between a flat plane and a globe (an arcminute is a sixtieth of a degree; the Moon is about thirty wide), and we have measured it.

Optics as stated: correctSite figures: correctHeight-dependence: real, and his sign is rightNaked-eye distortion: not there to seeShore-level shape test: settles nothing, either wayReflection angle at his site: refutes the flat plane
Where this lands

Everything Miller states about reflection is correct, his site figures are current, and his central expectation — that a globe compresses a reflection more as the observer rises — is true. At a drone’s 120 m ceiling a globe shortens the reflection by 4.0 per cent against 0.02 per cent for a flat plane, about 170 times apart, and refraction moves it by only a quarter of itself. His experiment would work; it is not something anyone can settle by looking.

The shore-level version — does the reflection look distorted — discriminates nothing in either direction. That question is about the shape of the mirror, and at eye height the mirror forming the mountain’s image is roughly 125 metres of water, not eighteen miles. Across 125 metres a globe and a flat plane differ by a third of a millimetre. That disposes of the claim; it equally disposes of the reply sometimes offered on our side, that a mirror-flat lake demonstrates curvature. It does not.

Where the reflection sits is a different and far larger quantity. The angle from the summit to its own reflection is twice the summit’s apparent elevation, the camera’s height cancels out of it, and the water supplies the level. Across every atmosphere Lake Pukaki could plausibly have had, a globe puts that angle at 311–317 arcminutes and a flat plane at 346–352. On a nature photograph taken there for its own sake, it measures 311 ± 3 arcminutes: inside the globe’s band and thirty-five arcminutes short of the flat plane’s. To reach the flat plane’s figure the air over the lake would have had to bend light upward along sixty-five kilometres of free troposphere, at about −170 K per kilometre — air that, from the 8.2 °C recorded at the lake, is past absolute zero before the ray is halfway up the mountain. The exclusion is not geometric, it is thermodynamic: what separates the models is the measured refractivity of air.

What he asks, and what he proposes

Q4, on p.47: “Do we observe perfectly mirrored reflections over large bodies of water at rest as we increase observer height?” The chapter on pp.113–116, “How optics are abused to fake curvature,” states the law of reflection correctly, shows the ephemeral lake at Badwater Basin mirroring the range beyond it, asks on p.115 “Are you able to produce a continuous, straight line of reflection from the Sun or Moon to the observer over a curved reflective surface?”, and on p.116 makes the proposal worth taking most seriously:

“Let’s find one of the longest reflective lakes with tall mountains in the background. Meet Lake Pukaki in New Zealand… In the near future, I will use computer-simulated environments to compare a globe to a topographic plane model. Which will more closely match what the drone captures at all observer heights: a curved or a flat reflective surface?”

With the expectation stated a sentence earlier: “If the Earth is a globe, we should see greater reflection distortion as the observer’s height increases.” That is a falsifiable experiment with a stated prediction, and it deserves an answer computed rather than asserted.

His site figures are right, and the site is well chosen

Aoraki/Mount Cook stands at 3,724 m, which is 12,218.5 ft — the post-2013 surveyed height from the University of Otago School of Surveying’s AORAKI2013 project, down from the 3,754 m set after the 1991 rockfall. His 12,218 ft is the current value, not a rounded legacy one. Lake Pukaki’s surface sits between 518.2 and 532 m depending on operating level, putting the summit around 3,200 m above the water. His “eighteen miles” is consistent with the 18.30 miles we measure from map coordinates. The shallow ephemeral lake at Badwater Basin in 2023–24 genuinely produced the reflections he shows. He also, without saying why, picked about the best site in the world for this; we come back to that at the end.

The answer to his question: yes, and here is how much

Solve the equal-angle condition on a sphere — find the specular point, the spot on the water where the ray from the summit reflects into the camera — and the reflection’s angular height follows at any observer height, under any refraction. For Aoraki at 64.6 km, on a globe and on a flat plane carrying the same air, Miller’s prediction comes out true.

Log-log chart of how much the reflection is shortened, in per cent, against observer height. The globe curve rises from 0.08 per cent at eye height to 4 per cent at 120 metres; the flat-plane curve stays near 0.02 per cent.
Reflection of Aoraki across Lake Pukaki against the direct view of the same mountain. On a globe the shortfall grows steeply with height; on a flat plane carrying identical air it barely moves. At a drone’s 120 m ceiling the two models differ by a factor of about 170.
observer heightglobe: reflection shortened byflat plane, same airon the visible mountain
1.6 m — standing on the shore0.08%0.000%0.1′
30 m1.36%0.006%1.6′
120 m — drone ceiling4.04%0.024%4.7′
400 m7.33%0.080%8.6′

Compression depends on the two features you measure between, and varies by more than a factor of two with that choice; these figures are for the visible upper mountain, 1,000 m above the water to the summit, subtending about 117′. Any drone measurement has to state its pair. Globe figures at k = 0.167, where k is the refraction coefficient: the fraction of the Earth’s curvature that bending in the air gives back, about 0.13–0.17 in ordinary air, zero for straight rays. On a globe the summit-to-reflection angle also creeps up with height, about +2′ by 120 m; on a flat plane it is height-independent.

The effect is real and his sign is right. A sphere is a convex mirror, convex mirrors minify, and the minification grows as the working area of the mirror moves away, which is what rising does. At eye height the specular point for the summit sits about 41 m in front of you; at 120 m it sits 2.5 km out. Refraction does not wreck it. Across the whole plausible band — a dry-adiabatic lapse (the steepest cooling with height that well-stirred air sustains) at k = 0.148 through a 20 K/km surface inversion at k = 0.329 — the 120 m compression runs 4.15% to 3.14%, a quarter of itself, against a factor of 170 between the models. But it is not visible, and it is not simple. Four per cent of a 117-arcminute mountain is 4.7 arcminutes, a sixth of the Moon’s width, on a reflection that is moving; and as a ratio it needs the drone’s height above the water to a few per cent, held framing, and both models built — substantial work for a signal that a different measurement at the same lake gives away.

Why the shape of the reflection shows nothing either way

At shore level the mirror doing the work is tiny. For an eye 2 m above the water, the visible upper mountain — the same 1,000 m-to-summit span the table uses — is assembled from a strip of water running from 41 m to 166 m in front of the observer (reflection_sim.py in the repository prints both endpoints). Roughly 125 metres of lake. The far shore never enters the optics, and neither does any part of the eighteen miles the argument is about.

Across those 125 metres a globe departs from a flat plane by 0.3 mm and the surface tilts end to end by 4 arcseconds (an arcsecond is a sixtieth of an arcminute), so light leaving it is redirected by twice that, 8 arcseconds. Miller himself takes one arcminute as the limit of sight; for that strip to bend the reflection by a single arcminute, its radius of curvature would have to be about 860 km, a globe seven times smaller than the one he is arguing against. So the mirror’s shape is a null in both directions; wind set-up and seiches (a lake’s slow slosh from end to end), centimetres over kilometres, are arcseconds across the strip that matters. One consequence of curvature that is real, though not measured here: on a globe the reflection is cut off at the bottom relative to the direct view, because the lowest strip of the mountain has no specular point this side of the horizon, whereas a flat plane truncates nothing.

The p.115 challenge resolves the same way. A glitter path is the statistical envelope of thousands of tilted wave facets, and those tilts are one to three orders of magnitude larger than anything curvature contributes. A continuous line of reflection over a curved surface is what Cox and Munk described in 1954 and what every satellite sun-glint correction assumes; the challenge asks for a surface nobody claims exists, a perfectly smooth ocean.

Log-scale comparison of the surface tilts that build a glitter path against the tilt contributed by Earth's curvature.
Every quantity that builds a glitter path is one to three orders of magnitude larger than the tilt curvature adds over the same patch of water; on a linear axis the curvature term would not be visible.

What to measure instead, at his lake, from his shore

Do not measure the reflection’s shape. Measure the angle from the summit down to the reflected summit.

Diagram: a camera at water level, rays to the summit on a globe and on a flat plane, and the mirrored rays below the waterline. The angle between a summit and its reflection is marked.
A ray arriving at elevation +e reflects off horizontal water to −e, so the angle between an object and its own reflection is twice the object’s apparent elevation. The camera’s height cancels out of it — to better than 0.2′ — so it never has to be known, and the water provides the horizontal reference without a level, a horizon, or an instrument.

That is why the measurement is available to anyone with a camera and a still morning: you need the peak’s height and distance, which are map work, and a frame containing the mountain and its reflection — not your own height, a horizon, or the far shore. The standard reply is that the map work is itself globe geodesy. It does not matter, because both models are handed the same inputs — the flat plane gets the same summit height and the same 64.6 km and is asked what angle it predicts. If those inputs were wrong, both predictions would move together, and the thirty-five arcminutes between them would not.

Curvature drops Aoraki 327 m over the 64.6 km to the camera, and ordinary air lifts about a sixth of that back. Doubled by the reflection, that is a thirty-five arcminute difference between the models — roughly ten per cent of the angle being measured, against four per cent for the drone’s compression at its ceiling.

The trap in this, which is the same trap the chapter falls into

Geometry alone cannot separate curvature from refraction — the two are exactly degenerate, and any argument that claims otherwise is wrong, ours included. A flat plane with refraction coefficient k′ = k − 1 is exactly degenerate with a globe at k. Both corrections go as D²/2R; we have checked the identity numerically at 20, 42, 56, 65 and 68 km and it holds to machine zero.

Chart of summit-to-reflection angle against refraction coefficient. The globe and flat-plane lines are parallel and offset by exactly one unit of k. The measured value crosses the globe line inside the physically possible band and the flat-plane line at k = minus 0.84.
The two model lines are parallel, offset by exactly 1.0 in k. The shaded band is the refraction real air produces: the measurement crosses the globe line inside it, at k = 0.16, and the flat-plane line at k = −0.84, exactly 1.00 apart and far outside it.

So the question is: what refraction would each model require, and can the atmosphere supply it? k = 0 is the autoconvective limit — the lapse rate at which air density stops decreasing with height and the atmosphere spontaneously overturns, about −34.2 K/km. Any negative k at all requires exceeding it, and this flat plane needs k = −0.84, about −170 K per kilometre. Follow that up the ray’s own climb from the 8.2 °C recorded at the lake: the air is below freezing 47 m above the water, liquefying nitrogen 1,179 m up, and past absolute zero at 1,626 m, with another 1,575 m of mountain still above it. That is not a cold atmosphere, it is an absent one. Super-autoconvective gradients do exist — they make desert mirages — but in centimetre-to-metre-thick layers over hot surfaces, and this ray path spends its life one to three kilometres above a cold winter lake.

The band is the day’s own: 8.2 °C and 956 hPa at the lake put a dry-adiabatic lapse at k = 0.148 and a 20 K/km inversion at 0.329. Better, the gradient can be measured: regressing temperature from ERA5 (the European weather centre’s reanalysis, a model constrained by observations) against elevation across twelve grid cells spanning 532 to 3,041 m, the height range the ray traverses, gives −6.65 K/km at R² = 0.986, a near-perfect straight-line fit — the ICAO standard lapse to within 0.15 K/km, a well-mixed winter airmass with no inversion anywhere in the column. It puts the day at k = 0.167; the reflection, read independently, implies k = 0.16. Those two did not have to agree. And the sign is the killer: the one strong anomaly real atmospheres produce at this scale is a surface inversion, the textbook product of a cold lake on a winter afternoon, and an inversion makes k more positive, pushing the flat plane’s prediction further from the measurement, not closer.

We ran it

The photograph is by Rach Stewart, taken at the SH8 pull-off near Lakestone Lodge on 5 August 2020 at 16:32 NZST, and used with her permission. She was making a nature photograph; the geometry in it is incidental, which is what makes it good evidence.

The Lake Pukaki photograph annotated with the identified summits, their bearings and distances, the mirror line, and the measured summit-to-reflection gap.
What the frame fixes before any earth model is chosen: peak identity, height and range from the map; horizontal scale from skyline features matched to their bearings, model-free because horizontal position depends on bearing alone. The bracket at right is the measured quantity, summit to reflected summit, 311′ ± 3′. Canon EOS 5D Mark IV · 70 mm · f/10 · 10 s · ISO 50 · 5 August 2020, 16:32 NZST. Photograph © Rach Stewart Photography, used with permission.

One thing would quietly decide the answer if skipped: the image scale has to come from the scene, not the file. A finished photograph that has been through Camera Raw, Photoshop and Lightroom carries cropping and, here, a 6.8 per cent vertical stretch — almost exactly enough to decide this question on its own. It is corrected for without guessing: horizontal scale from skyline features matched to surveyed bearings, vertical scale independently from the mountains’ surveyed relief, and the ratio comes out 1.068 ± 0.010 from two estimators that share no pixels. The skyline, the mirror line and the reflected summit are likewise measured rather than assumed (method notes).

The same photograph with the globe band, the flat-plane band and the measured position of the reflected summit marked. The globe band and the measurement overlap; the flat-plane band sits well below.
Both predictions and the measurement, on the frame. The globe’s two lines bracket the measured reflected summit: 311′ falls just inside 311′–317′. The flat plane’s pair sits a visible distance below. Photograph © Rach Stewart Photography, used with permission.
summit-to-reflection anglewhat it would require
Globe, every plausible atmosphere311′ – 317′k = 0.148 to 0.329
Flat plane, same atmospheres346′ – 352′k = 0.148 to 0.329
Measured311′ ± 3′globe: k = 0.16 — and the day’s measured k was 0.167
  flat plane: k = −0.84, or about −170 K/km sustained over 65 km
This photograph is consistent with a globe of the stated diameter under the conditions the site actually had, and it is inconsistent with a flat plane — not because of its geometry, which cannot decide, but because of the limits thermodynamics places on the refractivity of air, which can.

Why the shot has to be long, and why he chose well

The gap between the two models is D/R in angle — proportional to the distance to the mountain — and that governs whether the test is possible at all.

distance to the peakglobe-versus-flat-plane separation
5 km — a tarn with a hill behind it2.7′hopeless
29 km — the length of Lake Pukaki15.9′marginal
64.6 km — Aoraki from the pull-off34.8′comfortable

This is where Miller’s instinct to find “one of the longest reflective lakes with tall mountains in the background” earns its keep, though not for the reason he gives. The lake’s length is not what does the work; what the lake buys is a long, calm sightline, and what makes Pukaki exceptional is that Aoraki stands roughly 36 kilometres beyond the head of the lake, so the baseline is more than twice the lake’s length. Almost nowhere else combines a still-water mirror with a 3,700-metre peak at 65 kilometres. He found the best site in the world for this test and, we think, pointed the wrong instrument at it. One photograph is one photograph: what would extend or overturn it is below.

What would change our mind, and where this page could be wrong

Scope

This page answers Q4 and the reflection chapter. It does not address the wider bottom-up-disappearance material at pp.96–107, the Nikon P900/P1000 long-range photographs, or the power-line and tree-line images at pp.110–112, which are handled elsewhere on this site, and nothing here should be read as settling them.

Method notes

The mirror strip. The 41–166 m endpoints are the specular points for the summit and the 1,000 m contour at an eye 2 m above the water, solved exactly on the sphere (Alhazen’s problem) at the 60 km nominal range reflection_sim.py carries for the site; at the photographer’s 64.6 km the strip is a little longer and the sag across it still well under a millimetre. Sag and tilt are L²/8R and L/R for a 125 m chord on a 6,371 km sphere.

Skyline. Brightness confuses snow with cloud; colour does not. Terrain sits 40 to 70 levels from both sky and cloud in the red-minus-blue channel, so a threshold on that channel walks the entire two-thousand-column skyline without latching onto a cloud top.

Mirror line. The same colour test below the waterline recovers the reflected skyline. Paired column by column, the midpoint sits on a straight line to 3.3 pixels across 1,831 columns, verifying the reflection identity on the image; it also showed the frame rolled by about a quarter of a degree.

Reflected summit. Ten seconds of exposure smears the reflected skyline over about fifty pixels, and four defensible estimators applied to that edge disagree by nineteen arcminutes, more than half the signal. Instead the whole reflected massif is matched against the direct one across four hundred columns; two independent methods then agree to a single pixel.

Sources & further reading