The Mirror Settles Nothing. The Same Photograph Does.

Levi Miller asks whether perfectly mirrored reflections survive over large water as observer height increases, and proposes a way to find out: fly a drone at several heights over Lake Pukaki with Aoraki/Mount Cook in the background, then compare what it captures against computer models of a globe and of a topographic plane. It is a fair question, honestly posed, at a site he chose well. The answer to it is yes — the reflection really does compress as you rise, and his expectation has the right sign. But the effect is about four per cent at the legal ceiling for a drone, which is a thing you measure, not a thing you see.

The larger point is that he has picked the right lake for the wrong reason. The same site, in a single frame taken standing on the shore, carries a signal seven times bigger — and it is already sitting in photographs that tourists take there every week. It is not the shape of the reflection. It is the angle from the mountain down to its own reflected image. On this peak at this range a flat plane and a globe differ by thirty-five arcminutes in that angle, and we have now measured it.

What he asks, and what he proposes

Q4, on p.47, is one sentence: “Do we observe perfectly mirrored reflections over large bodies of water at rest as we increase observer height?” It is carried by the chapter on pp.113–116, headed “How optics are abused to fake curvature.”

The chapter states the law of reflection correctly and in two parts — equal angles, and everything in one plane — with a correct diagram. It shows the ephemeral lake at Badwater Basin returning a mirror image of the range beyond it. On p.115 it poses a challenge: “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 it makes a proposal, which is the part of the chapter we think is 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 plainly a paragraph earlier: “If the Earth is a globe, we should see greater reflection distortion as the observer’s height increases.”

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. Badwater Basin is a well-chosen illustration too: the shallow ephemeral lake that formed there in 2023–24 genuinely produced the reflections he shows.

He also, without saying why, picked about the best site in the world for this. That turns out to matter, and we come back to it at the end.

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

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

Compression is not a property of the mountain but of the two features you measure between, and it varies by more than a factor of two with that choice; these figures are for the visible upper mountain, from 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. Note that on a globe the summit-to-reflection angle also creeps up with height — about +2′ by 120 m, as the specular point walks out and the sag beyond it shrinks — where on a flat plane it is height-independent. The shore measurement is immune to this; a drone version would have to compute it.

Three things follow, and the first two are in his favour.

The effect is real and his sign is right. A sphere is a convex mirror; convex mirrors minify; the minification grows as the mirror’s working area moves further from the observer, which is exactly what rising does. At eye height the specular point for the summit sits 35 m in front of you; at 120 m it sits 2.5 km out.

Refraction does not wreck it. This is the objection we expected to have to make and it does not survive the arithmetic. Across the whole plausible band — a dry-adiabatic lapse 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 separating the models. His drone test is more robust than the shore-level arguments in the same chapter.

But it is not visible, and it is not simple. Four per cent of a 117-arcminute mountain is 4.7 arcminutes — about a sixth of the Moon’s width, on a reflection that is moving. Nobody adjudicates that by eye. Worse, it is a ratio, so running it properly means knowing the drone’s height above the water to a few per cent, holding the same framing, and building the two models to compare against. That last part is precisely what he proposes to do, and we would take the result seriously. We simply note that it is a substantial piece of work to extract a signal that a different measurement at the same lake hands over for free.

Why the shape of the reflection shows nothing either way

Before the better measurement, the null — and it is a null about the reflection’s shape, not about the photograph, which turns out to show a great deal. At shore level the mirror doing the work is tiny. Solving the same equal-angle condition for an eye 1.6 m above the water, the whole reflected massif — Aoraki’s summit down to the lowest ridge on the skyline — is assembled from a strip of water running from 35 m to 59 m in front of the observer. Twenty-four metres of lake. Even the faint reflected haze below the ridges is formed inside the first 150 m. The far shore never enters the optics at all, and neither does any part of the eighteen miles the argument is about.

Across the twenty-four metres that carry the mountain, a globe departs from a flat plane by 0.012 mm and the surface tilts end to end by 0.8 arcseconds. Take the generous version instead — the whole 150 m out to the faintest reflected haze — and it is 0.44 mm and 4.9 arcseconds, so light leaving it is redirected by twice that, 9.7 arcseconds. Miller himself takes one arcminute as the limit of sight. The requirement reads better backwards: for even that generous strip to bend the reflection by a single arcminute, its radius of curvature would have to be about 1,030 km — a globe six times smaller than the one he is arguing against. For the strip that actually carries the mountain, thirty-seven times smaller.

So the mirror’s shape is a null, and it is a null in both directions. It is not evidence for a flat plane. It is also not evidence for a globe, and we would ask anyone quoting a glassy lake back at this chapter to stop. One surface objection worth dismissing explicitly while we are here: sustained wind does tilt a lake’s mean surface, and a seiche sloshes it. Both are centimetres over kilometres — arcseconds across the tens of metres doing the work, and far below the mirror’s own noise floor.

The p.115 challenge resolves the same way. A glitter path is not a mirror image; it is the statistical envelope of thousands of individually tilted wave facets, and the tilts that build it are one to three orders of magnitude larger than anything curvature contributes. A continuous line of reflection over a curved surface is exactly what Cox and Munk described in 1954 and what every sun-glint correction in satellite remote sensing assumes. The challenge asks for a surface nobody claims exists: a perfectly smooth ocean.

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.

That is the whole trick, and it is why this measurement is available to anyone with a camera and a still morning. You do not need to know how high you are standing. You do not need a visible horizon. You do not need the far shore. You need the peak’s height and distance, which are map work, and a frame containing both the mountain and its reflection.

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, and available with your feet on the gravel.

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

There is a reason to be careful here, and it cuts against us as hard as against anyone. 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. No photograph of anything at any distance can separate curvature from refraction on geometry alone.

So the argument cannot be “the geometry proves a globe.” It has to be: given the geometry, what refraction would each model require, and can the atmosphere supply it? That question has a hard answer. 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, and the air is below freezing 47 m above the water, liquefying nitrogen 1,179 m up — a third of the climb — 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 are real, and they are what makes a desert mirage — but they live 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.

And the band is not generic either. The conditions at the lake that afternoon — 8.2 °C, 956 hPa — put a dry-adiabatic lapse at k = 0.148 and a 20 K/km inversion at 0.329, which is exactly the band used above, computed from the day’s own air. Better than that, the gradient itself can be measured rather than bracketed: regressing ERA5 temperature against model elevation across twelve grid cells spanning 532 to 3,041 m — the height range the ray actually traverses — gives −6.65 K/km at R² = 0.986. That is the ICAO standard lapse to within 0.15 K/km: an ordinary, well-mixed winter airmass with no inversion signature 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 refraction anomaly that real atmospheres do produce at this scale is a surface inversion — and a cold lake on a winter afternoon is the textbook site for one. An inversion makes k more positive, which pushes 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. Canon EOS 5D Mark IV, 70 mm, f/10, ten seconds at ISO 50. She was making a nature photograph; the geometry in it is incidental, which is exactly what makes it good evidence.

Four things had to be got right, and the first is the one that would quietly decide the answer if you skipped it.

The image scale has to come from the scene, not from the file. This is a finished nature photograph. It has been through Camera Raw, Photoshop and Lightroom, and like most such work it carries ordinary aesthetic cropping and a slight vertical stretch — here 6.8 per cent. That is not a defect; it is what makes it a photograph rather than an instrument reading. But it happens to be almost exactly large enough to decide this question on its own, so it has to be corrected for, and corrected for without guessing. We neither assume it away nor take it on trust: the horizontal scale is fixed by matching skyline features to their surveyed bearings, the vertical scale independently by the mountains’ own surveyed relief, and the ratio between them comes out 1.068 ± 0.010 from two estimators that share no pixels. Any measurement of this kind on a processed photograph has to do the same.

The skyline has to be found, not guessed. Brightness confuses snow with cloud. Colour does not: over this scene, terrain sits 40 to 70 levels away 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 once latching onto a cloud top.

The mirror line is measured, not assumed. Applying the same colour test below the waterline recovers the reflected skyline too. Pairing the two column by column, the midpoint sits on a straight line to 3.3 pixels across 1,831 columns — so the reflection identity is verified on the image rather than taken from a textbook. It also exposed a nuisance nobody had modelled: the frame is rolled by about a quarter of a degree, because nobody levels a tripod to better than that.

The reflected summit is not read off one edge. Ten seconds of exposure smears the reflected skyline over about fifty pixels, and four perfectly defensible estimators applied to that one edge disagree by nineteen arcminutes — more than half the entire signal. Instead the whole reflected massif is matched against the direct one across four hundred columns. Two independent methods then agree on the reflected summit to a single pixel.

What it reads

The measurement sits inside the globe’s band and thirty-four arcminutes short of the flat plane’s. Restated as the question the previous section says we have to ask: to put a flat plane on this photograph, the air over Lake Pukaki that afternoon must have had a lapse rate of about −170 K per kilometre, sustained along sixty-five kilometres of path spending most of its length one to three kilometres above the surface — an atmosphere that would be colder than absolute zero before the ray was halfway up the mountain, and carrying the opposite sign to the one a cold lake actually produces.

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

The gap between the two models in this measurement is D/R in angle — it grows in direct proportion to how far away the mountain is. That single fact governs whether the test is possible at all.

This is where Miller’s instinct to go looking for “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 — the mirror is 125 metres of water at your feet. What the lake buys is a long, calm, unobstructed 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 own length. Almost nowhere else combines a still-water mirror with a 3,700-metre peak at 65 kilometres.

He went looking for the best site in the world to test reflections against curvature, and he found it. We think he then pointed the wrong instrument at it — but the site is right, and the photographs that settle the question were already being taken there.

What would change our mind

• The original camera file. The vertical stretch is currently fitted from the scene rather than constructed from a known crop. The RAW, or simply the crop rectangle, would turn a measured 6.8 per cent into a derived one and retire the largest remaining uncertainty. The result moves in proportion to that correction, so if the file showed we had it wrong, the number moves with it — which is a reason to want the file, not a reason to doubt the measurement.
• The same measurement on more frames, from more photographers. One photograph is one photograph. The test costs nothing to repeat and the site is busy; a dozen frames from different days would turn a measurement into a distribution, and would expose any systematic in our pipeline that a single frame hides.
• A frame with a shorter exposure. Ten seconds is what smears the reflected edge. A fast frame would collapse the estimator spread from nineteen arcminutes to a few and remove the need for the whole-massif correlator.
• Miller’s drone experiment, run properly. With the height above water logged, framing held, and the two models built as he describes, it is a real test. If a flat plane matched the compression curve better than a globe across a range of heights, that would be evidence and we would say so.
• A sounding. The day’s lapse rate is now taken from reanalysis rather than assumed, which is a real improvement on a bracket — but it is a model field sampled across grid cells of differing elevation, not a vertical profile through the actual air. A radiosonde from that evening, or paired instrumented stations at different heights in the Mackenzie Basin, would replace it with a measurement. If one showed a strong inversion the reanalysis missed, the globe prediction moves up within its band and our agreement gets worse, not better.

Where this page could be wrong

• The vertical scale is the load-bearing number. It is measured to about one per cent by two independent estimators, and the quoted error spans them. But it is a fitted quantity on an edited file, and if it is wrong the conclusion is wrong in proportion. We would rather have the RAW.
• The lake level is now measured rather than assumed. Pukaki is a controlled hydro storage lake, and its level on 5 August 2020 was 522.889 m — Meridian’s own gauge record, audited by WSP and published by the Electricity Authority in its Hydrological Modelling Dataset, which carries a daily series for the lake from 1980 onward. That is 1.9 m below the figure we first assumed, and moves the prediction by 0.2′. The one loose end is the vertical datum: the source gives metres above sea level without naming the benchmark, though its stated control levels match Meridian’s published figures exactly.
• The visible summit is not the surveyed summit. The silhouette high point sits at bearing 358.62° against the geodetic 358.48° — ridge geometry, not error. It is why the measurement is made against the whole massif rather than against one named peak, and why two-point scale estimates from named summits are unsafe here.
• “Camera height cancels” is exact only to 0.2 arcminutes. The identity carries a small term in the camera height over the range. Immaterial against a 35-arcminute discriminant, but it is not literally exact, and it is worth saying so rather than rounding it to zero.
• There are three ways this measurement quietly fails, and each has to be guarded against. Assuming the file’s vertical and horizontal scales are equal; reading the reflected summit off a single smeared edge; and fitting the image scale against a synthetic skyline that was itself built on a globe. Any of the three yields a clean-looking answer that is not a measurement. The whole calculation was packaged and handed to an independent reviewer with those failure modes written up as the question rather than buried, and the figures above are what survived it. On the third, which is the one that sounds most damning: the horizontal scale is model-free by construction, since it comes from bearings alone and returns bit-identical values under either earth model; and the vertical fit was run under both, with the flat model free to choose any scale it liked. It fits the skyline four times worse in variance at its own best fit, and to do even that it wants 3.4 per cent of lens distortion — roughly ten times anything this lens does at 70 mm.
• Lens distortion is not modelled. The file has its lens profile switched off, so whatever the Tamron does at 70 mm is still in the frame. Two things keep it small: the summit and its reflection both sit near the vertical centre line, where a radial term vanishes on axis and is second-order just off it, and freeing a cubic radial term in the fit moves the scale ratio by 0.004. It is a real unmodelled term, and it is an order of magnitude below the signal.
• Everything here is computed for one specific viewpoint. The model figures are for the photographer’s actual position, 64.559 km from the summit. They do not transfer to another spot on the lake without recomputing: the globe-versus-flat-plane separation scales directly with distance, so a viewpoint a few kilometres nearer or further changes the numbers, though not the conclusion.

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. Nothing here depends on those pages, and nothing here should be read as settling them.

Sources & further reading

• Levi Miller, Globe Deconstruction? (prerelease) — Q4, p.47; reflection chapter, pp.113–116.
• Photograph: Rach Stewart Photography, Lake Pukaki, 5 August 2020, used with permission. https://www.rachstewartphotography.com/
• University of Otago School of Surveying, AORAKI2013 project — Aoraki/Mount Cook at 3,724 m by photogrammetry, GNSS and triangulation. https://www.otago.ac.nz/surveying/research/geodetic/aoraki2013-surveying-the-height-of-aoraki-mt-cook
• Lake Pukaki surface level on 5 August 2020 (522.889 m): Electricity Authority, Hydrological Modelling Dataset, 3_StorageAndSpill / 3_1_Storage / SI_PKI_Storage_LakePukaki.csv — a daily series from 1980, sourced from Meridian Energy’s own gauge and audited by WSP for the Authority. https://www.emi.ea.govt.nz/Environment/Datasets/HydrologicalModellingDataset
• Lake Pukaki operating range (518.0–532.5 m) and area: Meridian Energy. https://www.meridianenergy.co.nz/power-stations/lake-levels
• Site air temperature and pressure for 5 August 2020, 16:00 NZST (8.2 °C, 956 hPa): ECMWF ERA5 reanalysis via the Open-Meteo historical API. A model reanalysis constrained by observations, not a station reading; used only to bound the refraction coefficient.
• Terrain: Land Information New Zealand 8 m national DEM, queried via api.opentopodata.org (dataset nzdem8m); cross-checked against AWS elevation-tiles-prod terrarium tiles, which match it to 0 m median and 9.5 m RMS in this area.
• Refraction coefficient from the standard meteorological expression k = 503·(P/T²)·(0.0342 + dT/dh), reproducing the textbook 0.13–0.15 from a dry-adiabatic lapse. Autoconvective limit −34.2 K/km.
• C. Cox and W. Munk, ‘Measurement of the Roughness of the Sea Surface from Photographs of the Sun’s Glitter’, J. Opt. Soc. Am. 44(11), 838–850 (1954). Slope relations quoted from NASA’s ICESat-2 ATL13 Algorithm Theoretical Basis Document. https://icesat-2.gsfc.nasa.gov/sites/default/files/page_files/ICESat2_ATL13_ATBD_r005.pdf
• J. A. Shaw and J. H. Churnside, ‘Glittering Light on Water’, Optics & Photonics News 10(3), 43 (1999). https://www.montana.edu/jshaw/documents/Glitter_patterns_OPN1999.pdf
• ESA Sentinel-3 OLCI Technical Guide — 12.6° westward field-of-view tilt to mitigate sun glint. https://sentinel.esa.int/web/sentinel/technical-guides/sentinel-3-olci/olci-instrument/description
• Ibn al-Haytham, Book of Optics (c. 1021) — Alhazen’s problem, the reflection point on a spherical mirror.
• Our own scripts: spherical specular-point solve, compression ratios, colour-channel skyline extraction, whole-massif mirror correlator, scale and refraction fits. Packaged and handed to an independent reviewer; retained and publishable on request.
• Cross-references on this site: bottom-up observations (near-surface refraction machinery, thermometer protocol, refraction-coefficient convention); Rampion (long-path atmospheric cases, erect versus inverted images under inversions); the flat surface test (intact reflections used as a mirage diagnostic).