Visual Plane
every picture states the point it is correct from
Through water and glass

The sky inside a cone

From under water the whole sky — every direction out to the horizon — arrives inside a cone of 48.61°. Outside it the surface is a mirror. That cone is a picture surface, and it has a distortion no surface in the curved field has — an area scale that runs to zero.
18 min read 7 figures Not a projection at allThe surface is a choice

Look up from under water on a calm day and there is a bright disc overhead with a dark ring outside it. Inside the disc is everything above the surface — the sky, the clouds, the trees on the bank, someone standing at the edge. Outside it is the bottom of the pool, reflected.

The disc has a half-angle of 48.61° in fresh water, and it does not depend on the depth, the direction of view, or the size of the pool. It is arcsin⁡(1/n)\arcsin(1/n) and nothing else.

That number is already familiar from the first essay in this field, where it appeared as the critical angle — the largest angle at which light can leave water. Read the other way it is the largest angle at which light can arrive, and reading it that way turns the surface into something this site has a whole field for measuring: a picture surface.

Snell's window: the sky inside a cone of 48.6°Every direction above the water, right out to the horizon, arrives underneath within 48.61° of straight up. Beyond that the surface is a mirror. The rings are equal steps of the sky's own zenith angle; the last 5° before the horizon occupies 1 px of a 148 px radius.15°30°45°60°85°the rim: 48.61° from straight upbeyond it: the bottom, reflected45° of sky0.469460° of sky0.370275° of sky0.211485° of sky0.0738area scalen = 1.333, so the rim is at asin(1/n) = 48.61°area scale 0.563 at the centre, 0.0738 at 85°
Fig. 1 The window, drawn as a disc with the sky’s own zenith angles as rings. The rings are equal steps of 15°; the last 5° before the horizon occupies 1 px of a 148 px radius.

The map

A direction above the water at zenith angle θa\theta_a arrives underneath at

θw=arcsin⁡ ⁣(sin⁡θan)\theta_w = \arcsin\!\left(\frac{\sin\theta_a}{n}\right)

with the azimuth unchanged, because refraction at a plane keeps the ray in its plane of incidence. So the whole of the upper hemisphere, θa∈[0°,90°]\theta_a \in [0°, 90°], is compressed into θw∈[0°,48.61°]\theta_w \in [0°, 48.61°].

That is a map from directions to directions, which is exactly the shape of the objects in the curved field. The six surfaces there — plane, cylinder, stereographic, the two fisheyes, equirectangular — are each a rule for turning a direction into a mark, and each is measured on the three things a picture surface can do to the world: bend its straight lines, turn its right angles, and change its scale.

Snell’s window can be measured on the same three, and the results are worth having because one of them is unlike anything in that table.

Straightness

A great circle of the sky — the horizon itself, the path of the sun, any straight line at infinity — arrives underneath as a curve that is not a great circle, because the map compresses the zenith angle non-linearly. So the window bends straight lines, like every surface in the curved field except the flat plane.

That much is unremarkable. The interesting one is what happens at the rim.

Area scale, and the thing no other surface does

Differentiating the map gives two stretches. Radially,

dθwdθa=cos⁡θancos⁡θw\frac{d\theta_w}{d\theta_a} = \frac{\cos\theta_a}{n\cos\theta_w}

and tangentially, the ratio of the circles at the two zenith angles,

sin⁡θwsin⁡θa=1n\frac{\sin\theta_w}{\sin\theta_a} = \frac{1}{n}

exactly, at every angle — which is a small surprise on its own. The tangential stretch is a constant. All the variation is radial.

Their product is the area scale:

cos⁡θan2cos⁡θw\frac{\cos\theta_a}{n^2\cos\theta_w}

At the zenith it is 1/n2=0.5631/n^2 = 0.563: the sky directly overhead is compressed to just over half its solid angle. That is a number of the kind the curved field is full of.

At the horizon it is zero.

Both of the window's distortions, to the horizonThe area scale starts at 1/n² = 0.563 and goes to zero; the anisotropy starts at exactly 1 — the window is conformal on its axis — and goes to zero as well. Nothing in the curved field does either.00.2500.5000.7501020406080zenith angle of the direction above the water (degrees)area scale, and anisotropy — both 1 for a surface that does nothing1/n² = 0.563anisotropyarea scalen = 1.333at 89° the sky is compressed 135× in area
Fig. 2 Both stretches, all the way to the horizon. The area scale starts at 1/n² = 0.563 and goes to zero; the anisotropy starts at exactly 1 and goes to zero as well. Nothing in the curved field does either.

Zero area scale means infinite compression. The last few degrees of sky before the horizon — a band that in life contains the far bank, the trees, everything on the skyline — are squeezed into a sliver at the rim of the window whose width goes to nothing. The figure makes this a size rather than a claim: the last 5° occupies 1 px of a 148 px radius.

No surface in the curved field does this over the range it is drawn at. The plane’s width runs away toward 180° but stays finite everywhere it is defined; the cylinder’s area scale grows with sec⁡φ\sec\varphi; the equal-area fisheye’s is exactly 1 by construction. Snell’s window is the first surface on this site with an unbounded distortion inside its own domain.

Where the compression goes

It is worth doing the arithmetic on that sliver, because “infinite compression at the rim” is the kind of phrase that gets nodded at.

Take the band of sky between 85° and 90° from the zenith — the last five degrees, which in life is the whole skyline: the far bank of the lake, the treeline, the buildings, the horizon itself. Above water that band is 5° wide. Underneath, it arrives between 48.55° and 48.61°, a band 0.06° wide. It has been compressed by a factor of about eighty in the radial direction alone.

Between 89° and 90° — the last degree, which is where an object at any distance actually sits — the compression is a factor of several hundred, and it goes on increasing without limit as the horizon is approached. There is no angle at which it stops.

So everything on the horizon arrives at the rim of the window in a ring of essentially no width. That is not a defect of the figure’s drawing; it is the geometry, and it is why a diver looking up sees the surroundings of a pool as a bright rim rather than as a scene. The information is there and it is compressed past any usefulness.

The reason is visible in the closed form. The radial stretch carries cos⁡θa\cos\theta_a in its numerator, and cos⁡90°=0\cos 90° = 0. Every other surface on this site has a stretch that is a ratio of things that stay bounded over its own domain; this one has a factor that goes to zero exactly at the edge of its domain, which is the same thing as an area scale going to zero.

Conformal at the centre, and only there

The anisotropy — the ratio of the radial stretch to the tangential one — is

cos⁡θacos⁡θw\frac{\cos\theta_a}{\cos\theta_w}

which is exactly 1 at the zenith and falls to zero at the rim.

Being 1 at the centre means the window is conformal there: a small shape directly overhead arrives with its angles intact, merely shrunk by 1/n1/n in both directions. That is worth noting because stereographic is the only surface in the curved field that is conformal anywhere, and it is conformal everywhere. The window has a single conformal point.

Away from that point the anisotropy falls, which means shapes are squashed radially relative to their tangential extent. A circular cloud overhead arrives circular; the same cloud low in the sky arrives as an arc, flattened toward the rim. That is exactly what a photograph looking up from under water shows, and it is usually described as “the edge of the window is distorted” rather than measured.

The reverse view, which is the same map

Everything above is the sky seen from underneath. Turn it around and the same relation says something about looking into water from above, and the two readings are worth holding together because they are one function.

From above, the map runs the other way: a direction under water at θw\theta_w leaves at θa\theta_a with sin⁡θa=nsin⁡θw\sin\theta_a = n\sin\theta_w. So the cone of directions under water inside 48.61° expands to fill the whole hemisphere above, and directions under water outside that cone never leave at all — their light is totally internally reflected back down.

Which means: from above the surface, everything under the water is visible, and from below, everything above it is visible inside a cone. Not a contradiction, and not a violation of anything: the two statements are about different sets of directions, and light paths are reversible, so a ray that leaves the water at 80° from a direction 47° below is the same ray whichever way it is traversed.

The asymmetry that makes the two readings feel different is which hemisphere is being compressed into which. Looking down, the whole underwater hemisphere is not visible: only its inner 48.61°. Looking up, the whole above-water hemisphere is visible, compressed. Both statements are the same arcsin⁡\arcsin, and which of them sounds surprising depends on which side the observer is on.

The three numbers, checked twice

Every stretch above is available in closed form, and every one of them is also computed by differencing the map numerically. The two agree to a part in a million at every angle the figure draws, which is what says the differencing is measuring the surface rather than the closed form being a restatement of itself.

There is one place the differencing needs care and it is a good example of a check catching its own instrument. Both stretches are ratios of two quantities that vanish together at the zenith, so a central difference straddling zero asks the map for a direction below the water and refuses. The step has to be bounded by the distance to both ends of the range — and, separately, bounded below, because a step small enough to be swamped by cancellation reports a stretch that is mostly rounding noise. Getting that wrong the first time produced an anisotropy of zero at the zenith, which reads as a finding and was arithmetic.

The rim as a limit

The claim that the horizon lands exactly at the critical angle is checked as a limit rather than substituted: the window’s map is evaluated at θa=90°−10−6\theta_a = 90° - 10^{-6} and compared with arcsin⁡(1/n)\arcsin(1/n).

That is the same discipline the vanishing point of a direction gets on this site, and for the same reason. Substituting the closed form into itself proves nothing; approaching the value from inside the domain and finding the map converge to it is a statement about the map.

What lies outside the rim is the other half of the same relation. A ray arriving from beneath at more than 48.61° cannot have come from above — its refracted continuation would need sin⁡θa>1\sin\theta_a > 1 — so it must have come from below, by total internal reflection off the underside of the surface. That is the dark ring, and it is a mirror image of the bottom of the pool, complete and undistorted in the sense that reflection is a rigid operation.

So a diver looking up sees two pictures at once, joined at a circle: a compressed image of everything above the water inside 48.61°, and a reflected image of everything below it outside. The join is sharp because the transition is not gradual — beyond the critical angle the transmitted ray does not weaken, it ceases to exist.

What changes it

The window’s rim moves with the index, which the slider does. Sea water at 1.339 closes it very slightly, to 48.32°; a hypothetical liquid at 1.55 would close it to 40.2°. Fresh water’s 48.61° is the number worth remembering because it is the one that applies to almost every case anyone will meet.

What does not change it: depth, direction of view, the size of the body of water, the height of whatever is being looked at. Every one of those affects what is inside the window and none of them affects where the window ends.

Waves do, and this is where the model runs out honestly. Everything above assumes a flat surface. A rippled surface has a normal that varies from point to point, so the window’s rim is not a circle and the compression at the rim varies along it — which is why the effect is only clean on a still day, and why the crisp disc is a photograph of calm water rather than a diagram of it.

One room at 120° across, cast onto six picture surfacesEach panel holds the same angular width of the same scene. The flat plane keeps every straight line straight and pays for it at the edges; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces
Fig. 3 The curved field’s six surfaces, for comparison. Every one of them has a bounded area scale over the range it is drawn at, and this essay’s surface does not — which is the sharpest thing to say about it.

What is actually visible inside it

The window is a compression, and a compression preserves what is inside it — so it is worth saying plainly what a diver can and cannot read from the disc, since the answer is not “nothing”.

Angles near the middle survive. The window is conformal on its axis, so a shape directly overhead is faithful in form and reduced in size. Someone leaning over the surface directly above is recognisably themselves.

Positions do not translate linearly. Two objects 30° apart in the sky are not 30° apart in the window and are not the same fraction of the way out either, because the map is non-linear from the start: 15° arrives at 11.2°, 30° at 22.1°, 45° at 32.1°, 60° at 40.5°, 75° at 46.5°. Read that list once and the shape of the whole essay is in it — the steps start at three-quarters and shrink to a tenth.

Anything near the horizon is gone as information. Not dark, not hidden: present, and compressed past reading.

That list is the same one every picture surface in the curved field gets, and the reason it can be written at all is that the map is available in closed form. What makes this surface different from those is only the last line — every one of them keeps its horizon at a finite width, and this one does not, because no surface there has an edge to its own domain.

Straight, or conformal — the corner that is emptySix picture surfaces plotted by how much they bend a straight line (across) against how far they are from preserving shape (up). The plane sits on the left edge and the top; stereographic on the bottom and the right. Nothing sits in the bottom-left corner, and by Beltrami's theorem nothing can.straight AND conformal — empty10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10¹10²bend of a straight line, as a fraction of its own chorddeparture from conformal (degrees, or % of anisotropy)planecylinderstereographicequidistantequal-areaequirect.sampled over a 120° fanlower left would be a surface with no cost
Fig. 4 The curved field’s two measurements, for comparison. Every surface there trades straightness against conformality over a bounded range; the window’s stretches both run to zero, which is off the bottom of this comparison entirely.

Why it is not in the curved field

A reasonable objection: if Snell’s window is a picture surface, why does it sit in this field rather than beside the other six?

Because it is not a choice. The six surfaces in the curved field are decisions — a photographer or a cartographer picks one, and the essays there are about what each choice costs. Nobody picks Snell’s window. It is imposed by an interface, and the only decisions available are which side of the interface to stand on and what the liquid is.

That distinction matters more than it sounds. The curved field’s argument is that a picture surface is a decision with a measurable price and that no surface pays nothing. This field’s argument is that some maps are not projections at all, and the window is one of them: it is a map on directions, so it can be measured with the curved field’s instruments, but the picture a camera makes through the surface is not a projection through any centre, for the reasons two essays back.

Putting it here keeps that straight. The window is what the refraction field’s arithmetic becomes with the source pushed out to infinity and finite points set aside — which is the same simplification that made the slab’s vanishing points survive, applied to a case where the two media differ.

One more instrument, borrowed

The curved field measures its surfaces by differencing on the sphere rather than by evaluating closed forms, and there is a reason that is worth repeating here because the window is a good test of it.

Differencing means: take a small right angle at a point, push it through the map, and measure what comes out. The trap the expansion phase found is that the orientation of the right angle matters. Measuring the angle along the surface’s own coordinate directions gives the cylinder a perfect conformality score — 5.5 × 10⁻¹⁰ degrees, a flawless pass — because those two directions happen to stay perpendicular under the cylinder’s own map. Rotating the right angle through a half-turn and keeping the worst case rejects it by 5.62°.

Snell’s window has the same structure and would fail the same weak test the same way. Its radial and tangential directions stay perpendicular everywhere, so a test that only ever looks along them would report the window as conformal from the zenith to the rim. It is conformal at exactly one point, and the honest measurement — the ratio of the two stretches — says so.

That is the second time on this site that a necessary condition has been evaluated at the one input where it cannot fail, and the third counting the cross-ratio of four consecutive divisions. The pattern is consistent enough to be worth naming: a test evaluated in the coordinates of the thing being tested is usually not a test.

A flat port into water, and the field it takes awayRefraction at the port narrows every angle by Snell's law, so a 90° lens sees 64° underwater. No flat port, at any focal length, sees wider than 96.6° — twice the critical angle, and the ceiling is the same one Snell's window has.05010015050100150field of view in air (degrees)field of view behind a flat port, into water (degrees)no port at allthe ceiling: 96.6°114° → 78°90° → 64°63° → 46°46° → 34°28° → 21°n = 1.339a 90° lens becomes a 64° one
Fig. 5 The rim, read from the camera’s side. Nothing behind a flat window into water sees a field wider than twice the critical angle, and the curve approaches that ceiling rather than crossing it.
Apparent depth is a curve, not a numberDividing by n is correct looking straight down and nowhere else. At 80° from the vertical the bottom appears at 19% of its true depth rather than 75%.00.2000.4000.6000.800020406080angle of the line of sight from the vertical (degrees)apparent depth, as a fraction of the true depth1/n = 0.750n = 1.33375.0% straight down, 19.3% at 80°
Fig. 6 The same relation read the other way. Looking down into the water, the compression appears as an apparent depth that falls with the angle of view; looking up from under it, as a sky squeezed into a cone.
A line of sight 70° from the vertical, into waterThe ray bends by 25.2° at the surface, so an object 1.50 m down appears 0.543 m down — not the 1.125 m that dividing by n would give.water, n = 1.333eyetruly 1.50 m downappears 0.543 m downh/n would be 1.125 mno single viewpoint — the rays miss by 25.2° of bend at the surfaceapparent depth 36.2% of the true one, not 75.0%
Fig. 7 One ray of the window, drawn in section at a steep angle. Everything above is this picture with the source pushed out to infinity and the finite geometry set aside.

The ceiling it sets

One consequence to carry forward. Because the whole sky arrives inside 48.61°, no camera under water — behind a flat port, behind nothing, at any focal length — can photograph a field of the above-water world wider than twice that: 97.2°.

That is a hard ceiling and it is the subject of the next essay, where it turns out to have a companion result that is much less expected: there is a port shape that restores the pinhole camera exactly, and it costs nothing at all if it is centred to within a millimetre.

Where this goes next

plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces No picture surface keeps everything A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none. water, n = 1.333eyetruly 1.50 m downappears 0.818 m downh/n would be 1.125 mno single viewpoint — the rays miss by 17.1° of bend at the surfaceapparent depth 54.5% of the true one, not 75.0% What a ray does at a surface A pool looks three-quarters as deep as it is — but only if you look straight down. At sixty degrees the same bottom appears at half its depth, and at eighty at a fifth, which is why the far end of a pool looks shallow enough to walk in. 00.50011.50205101520the dome's centre, off the entrance pupil (mm)worst departure from the pinhole it would be in air (degrees)centred: exactly zero6 mm → 0.635°a 100 mm dome in acrylic, n = 1.4910.106° per mm of centring error The port that is not there A flat window into water costs a lens a third of its field. A sphere centred on the entrance pupil costs nothing at all — not nearly nothing, exactly nothing — and six millimetres off centre costs 0.635°.