Perspective Cannot Make a Sunset

1 · What the paper gets right, and we are not disputing

The core relation is φ(Z) = arctan(h/Z), with dφ/dZ = h/(Z² + h²) — the nonlinear mapping of depth into vertical visual angle. It is correct, it is uncontroversial, and it is the same equation our flat-surface page is built on: the 0.199′ depression of a waterline at 540 feet is this expression evaluated once.

Two things follow that the paper does not stress. First, this is a property of angular mapping, not of any defect: it is present identically in an ideal pinhole with no lens, no glass and no eye. Calling it “distortion” adds no physical effect at the object. Second, it is an effect on separations and backgrounds along the line of sight. It squeezes the gap between near and far things. It does not shrink a frontal object beyond the ordinary inverse-distance law, and it removes nothing.

2 · The equations are right. The sentences underneath them are not.

Paper 1’s fourth section is where the mathematics becomes conclusions, and it is the part the flat-earth application actually uses. Every equation in it is correctly written and correctly named — we say so in §1 and we mean it. Each is then followed by what the paper calls a Plain-English Translation — a sentence restating the maths for the reader. In each case that sentence says something the equation does not, and in three of the four it says the reverse. Here they are side by side. The paper draws none of them; the figures below are ours.

Quotations are from the copy of the PDF distributed at that link, read for this review. Three of the four responses are arithmetic on the paper’s own expressions and can be checked in a minute; the fourth is a standard result about geodesics on a sphere.

The rate is bounded. There is no spike near your feet.

Having correctly written dφ/dZ = h/(Z² + h²), its plain-English translation reads: when an object is near your feet, Z is close to zero, so the bottom of the fraction is small and the rate is “huge” — a “sudden, non-linear” effect that “physically curves the image path.”

The bottom of the fraction at Z = 0 is not zero. It is h². The rate has a maximum of exactly 1/h and is smooth everywhere — for eye height 1.5 m, 38.2° per metre at your toes, falling away steadily. Brisk, finite, and nothing sudden anywhere.

Higher objects sweep more slowly, not faster

The next subsection is headed “Dual Zenith-Nadir Symmetry”, and then argues that because H sits on top of the fraction, higher objects drop across the field of view much faster than lower objects rise. H is also in the denominator, squared. The peak rate is 1/H, so it is inversely proportional to height.

The dramatic plunge is a choice of starting point

The showcase comparison sets an overhead structure sweeping “a dramatic 90°” against a ground line managing “only a minor 56° that flattens almost instantly.” Both arithmetics are right. The gap between them is manufactured: the overhead case is started at Z = 0 and the ground case at Z = 1 m.

Start them at the same place and the asymmetry disappears, because arctan runs from 90° to 0° for every height. A ground point at Z = 0 is at the nadir, exactly as the aircraft at Z = 0 is at the zenith. Start both at one metre instead and the comparison reverses: the aircraft has moved 0.006°, the ground point 33.7°.

And the step the whole argument rests on is a chart artifact

The paper constructs the projection of a world point onto the unit sphere of the retina, correctly. Then it observes that dφ/dθ changes continuously along the image of a straight road, and concludes: a line whose slope continuously changes is by definition a curve.

Here is what its own construction says instead. A straight line in the world, together with the eye, defines exactly one plane through the centre of the sphere. That plane cuts the sphere in a great circle — which is the geodesic, the straightest available path on a sphere, with geodesic curvature identically zero. A straight line images onto a spherical retina as the straightest thing that surface permits.

And then the conclusion contradicts all four

Everything above converges on one paragraph, the summary the section is written to reach:

“Crucially, this non-Euclidean geometry applies symmetrically across the entire visual field. For objects located high above the observer’s eye level — such as cloud decks, airplane contrails, or power lines — the non-linear angular transformation forces these structures to descend at a significantly steeper visual angle than ground lines ascend, creating steep, downward-concave arcs that plunge aggressively to converge at the horizon vanishing point. Both biological eyes and camera lenses natively transform straight perspective lines into non-linear, curving arcs as they converge toward the optical horizon from above and below.”

A running header and page footer fall in the middle of that sentence in the PDF and are elided here; nothing else is changed.

There are four claims in it, and the paper has already refuted every one of them itself.

1. It contradicts itself inside one sentence. The geometry “applies symmetrically across the entire visual field” — and then, immediately, high objects descend “significantly steeper” than low ones ascend. Those cannot both hold. The first is the true one: arctan is symmetric, every height sweeps the same 90°, and the peak rate is 1/h whichever side of the eyeline you are on. The sentence states the correct result and then denies it in its second clause.
2. Steeper is backwards. This is the inverted derivative from the second row of the table. Peak rate is 1/H, so the higher the object, the slower it sweeps. Cloud decks and contrails are the slowest-moving things in the sky by this measure, not the fastest.
3. They do not plunge, and they do not converge. Both halves of that phrase are wrong, in opposite directions. The second derivative of arctan(H/Z) is positive, so the descent decelerates: for an aircraft at 9,144 m it falls 6.24° over the first kilometre and 0.02° over the kilometre at a hundred miles out. The arc flattens as it goes; it does not steepen. And it never arrives — “converge at the horizon vanishing point” is precisely what an asymptote does not do, as the paper’s own page 5 numbers show and as §7 below works through.
4. The last sentence is refuted by a theorem the paper states correctly. Elsewhere it writes that because X and Y are divided by Z at identical rates the ratio between them stays constant — which is the theorem that rectilinear projection maps straight lines to straight lines. So a camera lens does not natively turn straight lines into arcs; that is what it provably does not do, barring the radial term which is zero at frame centre. And the eye does not either: the image is a great circle, the geodesic. The paper proves both halves of its final sentence false in its own pages.

That single step decides the paper’s thesis, and it is a statement about coordinates rather than about light. Which is the pattern in all four: the mathematics is clean and the sentence attached to it is not, and the sentence is the part that gets quoted.

3 · Lens distortion is real, small, and exactly zero where every claim sits

Paper 1’s camera half rests on Brown–Conrady radial distortion — barrel and pincushion. Those are real: they bend straight lines by percent-level amounts at the edges of a frame. They also have a defining property that settles the matter. Radial distortion is a function of radius from the optical axis, so it vanishes on the axis.

Every observation this paper is cited for — the setting Sun, Polaris, the raft on the pond — is a centred subject. The photographer points at the thing. At frame centre the radial term is zero by construction, and on a phone it has usually been corrected digitally as well. Whatever barrel distortion does at the corner of an image, it cannot lower, truncate or hide a subject in the middle of one.

The paper agrees, in a table it prints as evidence for the opposite. Its measured-deflection table runs down the frame and ends at a row labelled Vanishing Pt, whose deflection column reads 0 px. By its own tabulated numbers the lens does nothing at all at the horizon — which is the one place every claim it is cited for actually happens.

We are holding paper 1 to its printed numbers there, not vouching for them. Where that table came from is a separate question and a bad one for it — taken up on the companion page — but it does not need settling here. The row is in the document, presented as evidence, and it reads zero.

The cited literature says something smaller still, and it is worth reading because it is short and public. Brown–Conrady is a real model and a fair name to use — it covers the radial terms and the decentering ones together. But the 1966 paper paper 1 points at is the decentering half. It takes symmetric radial distortion as already handled and works on what is left when a lens’s elements sit slightly out of true, and Brown’s opening paragraph sizes that residue: holding it under five microns across the plate takes “appreciable skill and patience”, and getting under two microns “calls for luck in addition to skill and patience.” His plates are 190 × 215 mm. That is an effect of a couple of parts in a hundred thousand of the frame — small enough that detecting it at all is what the paper is for. What paper 1 needs from a lens is the removal of the lower half of a ship.

This is worth stating gently, because in the optics debate Miller appears to name barrel and pincushion as the correct technical categories and to distinguish them from what he means by “linear perspective distortion.” That is an accurate distinction and he is right to draw it. The difficulty is that the paper’s camera-side case is built on the very terms he sets aside. We take that from a condensed summary of that video rather than from its audio, and the working notes on it flag several segments where the two speakers’ labels are transposed. The distinction belongs to the sceptical side of the exchange, which is his; we have not confirmed the wording or the timestamp, and it should be checked before anyone quotes it.

4 · The eye half and the camera half share no mechanism

Roughly half of paper 1 concerns perceptual space: Luneburg’s hyperbolic model of visual space, the Hillebrand and Blumenfeld alley experiments, vertical-horizon shifts, empty-field myopia, oculomotor micropsia. These are claims about how the brain represents space. Their standing is a separate question — Luneburg’s model is contested and largely superseded — and we do not need to settle it.

The alley experiments are worth showing, because paper 1 describes them at length and does not draw them, and because what they actually were bears on what they can be used for. An observer sits in a dark room. Two rows of small flames — later, rods — recede into the distance in front of them. The furthest pair is fixed. Every nearer pair can be slid sideways and nothing else. Then the observer is given one of two instructions.

The two alleys do not coincide, and that is the whole finding. Asked to make the rows look like parallel straight lines, observers set them very nearly straight. Asked instead to make every pair’s gap look equal to the far pair’s gap, they set a visibly wider figure that bows outward. Since physically parallel lines have constant separation, a Euclidean visual space would give the same answer to both instructions. Drawn from the description in Erkelens (2015); the separation between the three is exaggerated here for legibility.

Luneburg read that discrepancy as evidence that visual space has constant negative curvature — that it is hyperbolic rather than Euclidean. Whether that is the right reading has been argued over for seventy years and we do not need to settle it. Two things about the experiment settle the use it is being put to.

It is a binocular judgement, and there is nothing for a camera to do here. The observer is reporting how a configuration looks to two eyes and a visual cortex. There is no photograph in it, no lens, and no measurement of where anything actually is — the whole datum is where a person chose to put a flame so that it satisfied an instruction about appearance.

And the effect runs the wrong way for the argument. Stated as a behaviour rather than a geometry, what the alley experiments found is this: people place the objects nearest them further outside parallel, and that tendency fades as the objects get further away. The departure from Euclidean expectation is largest at the observer’s feet and smallest at the far end, which is why both alleys close back onto the fixed far pair.

And there is a version of this that needs no interpretation at all. The monograph attaches a formula to those experiments: X(Z) = X₀ cosh(σZ). A hyperbolic cosine is smallest at Z = 0 and grows from there, so that formula describes an alley which is narrowest at the observer and widens with distance — the exact reverse of the settings the experiments recorded, and of the diagram above. The formula it cites as the quantitative result of the alley studies contradicts the alley studies, and you can see it from the formula alone.

That is the opposite shape of what the argument needs. To hide a ship at eight miles, or to bring a sun down at three thousand, a mechanism has to get stronger with distance — negligible nearby, decisive far off. The alley result is negligible far off and measurable nearby. Whatever it demonstrates about visual space, it cannot be the thing that grows into a horizon, and the paper does not notice that its own headline evidence is pointing backwards.

It is also the experiment the lead monograph describes as having used twenty-four green light-emitting diodes on motorised transverse tracks, in 1902 and 1913. The apparatus was flames and rods, and light-emitting diodes would not exist for another half-century. It is also the researcher the monograph calls “Steel Blumenfeld”, four lines above a reference list that correctly gives Blumenfeld, W.

We do not need to because the observations at issue are camera observations. A sensor has no visual cortex. Whatever the eye does with binocular alley judgements has no route into a stack of exposures from a tripod. The document places the perceptual material and the camera material side by side and the reader carries an impression from one to the other, but no mechanism crosses between them.

5 · Nothing in 52 pages derives an object being hidden

Take the entire argument as granted for a moment. What it delivers is a claim about the shape of the projected path of a straight line across an image surface — that lines which are straight in the world can render as curves. Suppose that is so. A curve is not an occlusion. Nothing in the document derives an object being cut off, truncated, or displaced below a boundary, and nothing derives the crisp lower edge that every horizon photograph shows.

There is a sharper way to put it, and it is the same test that decides the pond. The two mechanisms in play fail in different shapes, and the difference is checkable in one frame.

A resolution limit takes detail away everywhere at once. It has no favoured direction: the object softens on every side, its contrast collapses, and it goes out from the outside in, shrinking toward a point. Nothing about that produces a clean horizontal edge with a sharp top above it.

Angular compression does have a direction — and it still cannot hide anything. It squeezes vertical separations far harder than horizontal ones, so it genuinely is directional, and it would be wrong to pretend otherwise. But arctan is strictly monotone: it maps higher points to higher points, always, without exception. It can crush the gap between a hull and a waterline toward nothing, and it can never reorder them, never close the gap entirely, and never carry the hull to the far side of the horizon line. Compression makes things small. It does not make them absent.

Which gives the test. A compressed hull is still in the frame, occupying fewer pixels than it deserves — so magnification brings it back. An occluded hull is not in the frame at all, and no amount of zoom returns it, because there is nothing there to enlarge. Instrument-limited losses are recoverable by a better instrument; geometric ones are not.

So the two accounts leave opposite signatures and a single frame separates them: a ship with a clean waterline and a sharp superstructure above it is being blocked by something, a ship fading into a smudge is unresolved, and a long lens tells you which. That is the point on which the pond footage turns.

The Euclid-style treatise inside the same PDF is more careful about this than the lead monograph is. Its distance theorems say that objects become indistinct with distance — a statement about detection, which is compatible with the detection-floor analysis on our bottom-up observations and flat-surface test pages, and which we largely agree with.

6 · How the book uses it, and what its own picture says

Paper 1 is the theory; the book is where it is applied to the sky. At p. 128 the argument opens like this:

“The trailing path of an airplane that flies straight overhead appears to trace a spherical arc toward the horizon due to perspective. Before reading this book, you may have mistaken this visual phenomenon for the airplane moving around the physical curvature… Since the community has conducted so many long-distance observations at this point, we now know that the bottom of boats disappearing is due to angular-resolution limitations in the water. In this chapter, we will apply those same rules of angular resolution and linear perspective to the sky. By the end, you will realize that the globe defenders have no evidence to defend their light-year distances to the stars.”

The first sentence is correct and we concede it without qualification. A contrail laid down in a straight line at constant altitude does appear to trace an arc across the sky, and the reason is perspective, exactly as he says. It is the same result as §2’s fourth figure: the trail and your eye define a plane, that plane cuts the sky in a great circle, and a great circle drawn across a hemisphere looks like an arc. Nobody on our side attributes an overhead contrail’s arc to the curvature of the Earth, and anyone who did would be wrong.

But notice what the concession is worth, because it is worth nothing to either side. Ask the obvious question: would a contrail arc if the Earth were flat? It would. A straight trail and your eye define a plane; the plane cuts the sky in a great circle; a great circle across a hemisphere reads as an arc. Every step of that is the projective geometry of a straight line in ordinary Euclidean space, and the shape of the ground appears nowhere in it. The arc is what a straight line overhead looks like, full stop.

So this is a degenerate observation — both models predict it, and it therefore separates them not at all. It is the same structure this review keeps meeting: an appearance that follows from the geometry of looking, presented as though it favoured one arrangement of the world.

There is a wrinkle here that runs mildly against us, and we would rather state it than have it found. On a globe an aircraft holding a constant altitude along a great-circle track is not flying a straight line in three dimensions at all — it is following a circle of radius R + H — so its trail images as something very slightly off a great circle. On a flat plane the trail is genuinely straight and the image is a great circle exactly. The perfectly geodesic case is his, not ours. The departure is small and easy to size: a trail 200 km long following a circle of radius R + H stands about 784 metres proud of the straight chord joining its ends, which is a fraction of a degree seen from the ground and far below what the chapter is arguing from. But the exactness belongs on his side and the page should say so.

What follows does not come with it. That arc is a geodesic, it is visible along its whole length, and nothing anywhere on it is hidden. Deriving the disappearance of a ship’s hull from it — and then the distances to the stars — requires each step to be argued separately, and the paper the chapter rests on does not supply them.

This is the asymptote argument in visual form, and the next section works it in numbers.

7 · The asymptote, which is the paper’s own arithmetic

Paper 1’s showcase example on its page 5 is an aircraft at 30,000 feet. It computes the elevation angle as the aircraft recedes: 90° overhead, 48.8° at 5 miles, 10.8° at 30 miles, 3.2° at 100 miles. That is φ = arctan(H/Z), worked correctly.

Now read what it says. The angle is positive at every finite distance. It decreases without bound but never reaches zero, never goes negative, and at no point does the object’s lower limb vanish while its upper limb remains. On this mathematics an elevated body approaches the horizon and stays above it forever.

A sunset is not that. A sunset is a disc meeting a line, its lower half going out first, the remaining half keeping a hard edge, and then nothing. A Sun at fixed height above a plane, subject to every mechanism in this paper, can be made dimmer, smaller and blurrier as it recedes. It cannot be made to set.

None of this needed to wait for the horizon

The discriminator between a receding lamp and a distant Sun is angular size, and angular size is most sensitive exactly where the argument is least often made — high in the sky, close to the observer, hours before anything approaches setting. This is the most portable test on the page: it needs no horizon, no sea, and no argument about what happens at a limit. It can be run at ten in the morning.

The reason is that a lamp at height h seen at elevation α is at distance h/sin α, so its angular diameter goes as sin α. The height cancels. There is no free parameter to tune, and it does not matter whether the lamp is three thousand miles up or thirty:

By mid-morning the Sun should be visibly smaller than it was at noon. By the time it is a third of the way up the sky it should be half the diameter it had overhead — not subtly, not arguably, but obviously, in any filtered photograph taken on the same lens an hour apart. What is measured instead is a diameter constant to a fraction of a per cent all day, which is the behaviour of something so far away that a few thousand miles of sideways travel changes nothing. Our companion page on the Sun’s angular size works this through with the measurements, the historical record and the two-frame test anyone can run; we do not repeat it here.

That is the part worth holding onto. Everything below concerns what happens at the horizon, and the horizon is where this argument is always conducted — but the mechanism has already failed by mid-morning, at high elevation, where its predicted effect is largest and most easily checked. Whatever perspective does at a vanishing point, it never got the chance.

And the endpoint is not the only thing the mechanism has to reproduce. It has to produce the path, and the path is where this gets much harder.

The descent has to be manufactured out of horizontal motion. A lamp above a plane moving away from you recedes horizontally; nothing about it goes down. Every bit of apparent downward motion has to be supplied by perspective, dragging the image toward a vanishing point on the horizon along the lamp’s bearing. That gives a particular shape of descent — a fall toward one point, steepening as the lamp goes, with the disc shrinking all the way. The Sun we actually watch does something else. It keeps a constant angular diameter as it goes, and it travels along a smooth circular arc inclined to the horizon, moving sideways as much as it moves down. It does not fall toward a point. It slides along a track.

And the tilt of that track is your latitude. The setting Sun does not descend vertically except on the equator. Its path meets the horizon at an angle of 90° minus your latitude — exactly so at the equinoxes, within a few degrees of it through the rest of the year — 90° at the equator, 52° at San Francisco, 38.5° in London, 30° at Oslo. Perspective has no term for where the observer is standing. Pulling an image toward a vanishing point produces the same geometry for everyone facing the same way, so a perspective account owes an explanation of why the lean varies smoothly with latitude and why it equals that particular angle.

Then it has to reverse across the equator. Face west at sunset in the northern hemisphere and the Sun comes down from your left, leaning down-and-to-the-right, because it has crossed the southern sky. Face west at sunset in Sydney and it comes down from your right, leaning down-and-to-the-left, because it crossed the northern sky. Same direction of view, same setting Sun, mirrored path — and at 51.5° north and 51.5° south, mirrored by exactly the same angle. A vanishing point does not know which side of the equator you are on. A rotating sphere with an axis does, and that is the whole of the explanation on our side: you are on a ball that turns, and which way the sky appears to wheel depends on which half of it you are standing on.

One fallback deserves closing by name, because it is where this argument usually retreats. If perspective cannot lower the Sun, perhaps its lower limb merely merges into the compressed band of sky just above the horizon. It cannot: a half-degree disc cannot conceal itself inside a compression zone thinner than half a degree while remaining half a degree wide. The Sun that touches the horizon is the same angular size as the Sun at noon, and anything that hid part of it would have to shrink it too.

Polaris makes the same point from the other side. If elevation followed arctan(H/Z), the star would approach 0° only as distance ran to infinity, and would remain above the horizon everywhere on an unbounded plane. What is observed is that its altitude equals the observer’s latitude, falls linearly at about one degree per 69 miles, reaches exactly 0° at the equator — a finite, reachable, chartable place — and is simply absent south of it. A linear law terminating at a finite distance is the signature of a rotated horizon on a curved surface. It is not the signature of an asymptote.

8 · Where the document came from

Everything above is a physics response and stands on its own. Separately, the PDF itself raises questions of authorship and sourcing that a reader may reasonably want answered, and some of what we found there is serious. We have put that on its own page deliberately, so that the physics is not read as resting on it, and so that the sourcing findings can carry their own verification status. That page is still in draft. Each of its findings prints its own confirmation state, and nothing there should be repeated without the caveat attached to the line it comes from. Nothing on this page depends on any of it.

The short version: the file is not one paper but five documents concatenated, with different authorship and very different standards, and the lead monograph — the one actually cited — carries several indicators of machine generation with invented experimental detail. That is a claim about a document, not about Miller, who cites it in good faith as a source rather than presenting it as his own work.

What would change our mind

1. Derive an occlusion. Show, from any mechanism in the paper, an object whose lower limb is removed while its upper limb stays sharp, at a stated distance and height. Curvature of a projected line is not this, and detection failure is not this.
2. Break the asymptote. Show a distance at which arctan(H/Z) reaches or crosses zero for finite Z and finite H.
3. Show a centre-frame radial term. Demonstrate barrel or pincushion distortion displacing a subject on the optical axis by a measurable amount, on any lens.
4. A sun that never passes overhead. On a flat plane with a circling lamp, no observer is ever directly beneath it except those on its track; on a globe, everywhere between the tropics gets the Sun at the exact zenith twice a year, on dates fixed by latitude, and people photograph it (shadows vanishing at the foot of a vertical pole). Show that those zenith passes do not occur where and when the sphere puts them.
5. Polaris without a curved surface. Produce a flat arrangement giving altitude falling linearly with distance and terminating at 0° at a finite, identifiable line, rather than approaching zero asymptotically.

Where this page could be wrong

We would rather flag these ourselves than have them found.

We are answering the lead monograph, not all 52 pages. The anthology behind it contains careful Euclidean geometry which we have not reviewed theorem by theorem, and which is in places correct and simply over-extended.

We have not resolved the perceptual-space literature. We argue it is irrelevant to camera claims rather than that it is wrong, and that argument would fail for any observation actually made by eye without an instrument.

The page 5 numbers are read from the document. They are the paper’s own worked example as printed; we recomputed arctan(H/Z) and reproduce them, but we did not re-derive the aircraft scenario’s assumptions.

Sources & further reading

• “Visual Geometry is distorted compared to Tangible Geometry”, 52 pp., linked from shapedebate.com/42 and cited in the descriptions of the raft video and the optics-debate video. Our copy is the one distributed at that link. Structure, authorship and sourcing are examined on the companion page.
• W. Blumenfeld, ‘Untersuchungen über die scheinbare Grösse im Sehraume’, Zeitschrift für Psychologie 65 (1913), 241–404, and F. Hillebrand, ‘Theorie der scheinbaren Grösse bei binocularem Sehen’ (1902) — the alley experiments, as cited in the modern literature (Erkelens, ‘The Perspective Structure of Visual Space’, i-Perception 2015, doi:10.1177/2041669515613672). Establishes the standard form of both citations.
• R. Hartley & A. Zisserman, Multiple View Geometry in Computer Vision (2nd edn, Cambridge, 2004) — the standard reference for the result §2 and §3 both lean on, that rectilinear pinhole projection maps straight world lines to straight image lines. Cited for a textbook result rather than pulled for this page.
• D. C. Brown, ‘Decentering Distortion of Lenses’, Photogrammetric Engineering Vol. 32 No. 3 (May 1966), 444–462 — the source behind the Brown–Conrady model paper 1 invokes, and the paper it credits its measurement table to. Read in full: a glass-plate stellar calibration, measured by hand on a comparator, reported in microns, with two tables in nineteen pages — and treating the decentering residue rather than the symmetric radial barrel and pincushion paper 1 names. The companion page gives the detail.
• Companion pages on this site: the flat-surface test and bottom-up observations, where the same compression equation is granted and the detection-floor account is set out; and celestial globes, which answers the Polaris form of the argument as the book states it.