The port that is not there

Almost everything in this field is a loss. Water destroys the invariant; a slab moves every finite point; a refracted picture has no station point at all. This essay has one of each: a loss that is larger than most people expect, and a repair that is more complete than anyone would guess.

The loss is what a flat window does to a lens’s field of view. The repair is a piece of glass with a shape chosen so that Snell’s law has nothing to act on.

The flat port, and the third it takes #

Put a lens behind a flat window with water on the far side. A ray leaving the pinhole at $\theta$ in air arrives in the water at $\theta_w$ with $\sin\theta = n\sin\theta_w$, so the cone the lens covers is narrowed by exactly the relation from the first essay in this field.

The arithmetic is unkind. A 90° lens — a fairly wide one, 20 mm on full frame — sees 64° under water. A 114° lens sees 76°. The loss is not a fixed number of degrees; it is a compression that gets proportionally worse the wider the lens starts.

The ceiling is the interesting part of that figure. As the lens in air gets wider and wider, its underwater field does not go on growing — it approaches twice the critical angle and stops. No flat port, behind any lens, at any focal length, can photograph a field wider than 97.2° in fresh water.

That is the same number as the rim of Snell’s window, and it is the same fact stated from the camera’s side rather than the diver’s. There is no light in the water at more than 48.61° from the port’s normal that came from in front of the port; there is nothing there to photograph. A fisheye behind a flat port is a fisheye photographing a 97° world.

The magnification everyone quotes #

The usual way the flat-port loss is stated is as a magnification: things look a third bigger and a quarter closer under water, and the factor quoted is 4/3, which is $n$.

That is the small-angle version of the curve above, and it is exact on the optical axis and nowhere else — the same relationship the apparent-depth number has to its own curve. Near the axis, $\sin\theta \approx \theta$, so $\theta_w \approx \theta/n$ and the image is magnified by $n$. Away from the axis the sines matter and the magnification falls, which is why the curve in the figure bends away from a straight line through the origin.

So the honest statement of the flat-port effect has the same two-part shape as the apparent-depth one: a factor of $n$ on the axis, falling continuously to a ceiling at the rim. Anybody who has photographed a rectangular subject through a flat port has seen the consequence — the middle magnifies by 4/3 and the corners by less, which reads as a pincushion distortion and is nothing of the sort.

That last sentence is worth dwelling on, because it is the field’s recurring warning in a new place. The pattern looks radial. It is not a function of the image point, so it is not a lens distortion, and fitting it as one produces a coefficient that is correct at one subject distance and wrong at every other.

Why a flat port also distorts #

Losing field is the visible cost and it is not the expensive one.

The interface sits at a finite distance from the entrance pupil, and the displacement it introduces depends on how far the object is beyond it — the general result of this field. So a flat port is not merely a focal-length multiplier. It is a map that is not a function of the image point, so it is not correctable by any warp of the image, and the picture it makes has no station point.

That is why an underwater photograph behind a flat port has corners that look soft and stretched in a way that resists correction, and why underwater photogrammetry models the interface rather than fitting a distortion polynomial. The polynomial fits, at one distance. It is wrong at another, in a way a real lens never is.

The shape that costs nothing #

Now the repair, and it is a genuinely elegant one.

Snell’s law bends a ray by an amount that depends on the angle between the ray and the surface normal. At normal incidence — the ray along the normal — the bend is exactly zero, at any index, for any pair of media.

So: make the interface a sphere, and put its centre at the camera’s entrance pupil. Every ray of the picture passes through the pupil, and every ray through the centre of a sphere meets that sphere along its own radius, which is the normal. Every ray, at every angle, at normal incidence. Snell’s law has nothing to act on.

The camera behind such a port is the camera it was in air: same focal length, same field of view, same projection through the same centre. Not corrected — unchanged.

The figure checks this rather than repeating it, because a claim of exactness deserves a measurement. Over fourteen rays spanning the field, the worst departure from the entry angle is 7e-15°, which is double-precision zero and not a small residual.

The measurement that makes it a claim #

The second half of the figure is what turns the first half into something rather than a tautology.

A check that passes at every offset is not checking the centring. So the same computation runs with the sphere’s centre moved off the pupil, and the departure has to appear: at 6 mm it is 0.635°, and the cost is 0.106° per millimetre of centring error.

The curve of that departure against offset starts at exactly zero and rises smoothly, which is worth noting because it says the tolerance is a tolerance rather than a cliff. A dome centred to within a millimetre or two is very nearly a pinhole; one centred to within a centimetre is noticeably not.

That is why underwater housings for wide lenses come with extension rings of specific lengths and tables saying which ring goes with which lens. The ring’s job is to put the dome’s centre of curvature on the lens’s entrance pupil, and the tables exist because the pupil’s position is a property of the optical design that no dimension on the outside of the lens reveals.

Where the pupil is, and why it is hard to find #

The entrance pupil is the point every ray of the picture appears to pass through, seen from in front of the lens. It is the image of the physical aperture stop formed by the elements ahead of it, so it is not the aperture, not the front element, not the sensor, and not usually marked anywhere on the barrel.

It moves with focal length on a zoom. It moves, slightly, with focus. On some retrofocus wide-angle designs it sits well forward of the front element, in the air in front of the lens, which is disconcerting and perfectly ordinary.

There is one thing that finds it without opening the lens: rotate the camera about a candidate point and see whether near and far objects stay in registration. At the pupil they do; anywhere else they do not, and the misregistration is a parallax that falls as one over the distance. That is the subject of the last essay in the lens field, and it is the same quantity this essay needs — which is why the two fields meet here.

So the practical procedure for centring a dome port is: find the entrance pupil by the parallax test in air, then choose the extension ring that puts the dome’s centre there. Two measurements, both geometric, neither requiring anything to be taken apart.

Two ports, side by side, in numbers #

It is worth having the comparison in one place, at one configuration: a 90° lens in fresh water.

Only the last two rows are geometry rather than photography, and they are the ones this site is about. A flat port produces a picture that is not a projection of anything from anywhere; a centred dome produces a projection through the same centre the camera had in air.

The middle row of the table is the one that decides how much effort a measurement is going to take. A departure that grows with subject distance is the one thing an image-space correction cannot handle, at any order of polynomial, because the polynomial is a function of the image point and the departure is not.

What the dome does not fix #

Three honest limits, because “restores the pinhole exactly” invites more confidence than it deserves.

It is exact for a concentric shell only. A dome has a thickness. The inner and outer surfaces are two spheres, and if they share a centre then the second refraction undoes the first exactly and the whole shell is transparent to direction. If they do not — a moulded dome with a thickness that varies, or one deformed by pressure — the cancellation is incomplete. This is why dome ports are made to a tolerance rather than merely to a radius.

It moves the focus. A concentric dome forms a virtual image of the underwater world at a distance related to the dome’s radius, so the lens has to focus much closer than the subject actually is. That is a photographic problem rather than a geometric one, and it is why dome ports are used with lenses that focus close and often with a supplementary close-up element. Nothing here is affected: the directions are unchanged whether or not the lens can focus on them.

It is still under water. Everything about the medium — absorption, scattering, the colour shift — is untouched by the port’s shape. The geometry is restored and the photography is not made easy.

The parallax test, in more detail #

Since the whole dome argument rests on knowing where the entrance pupil is, and since nothing on the lens says, the test deserves stating properly.

Set the camera on a rail that lets it slide back and forth along its own axis. Find two objects at very different distances that line up when seen from the starting position — a near post and a distant chimney, say. Rotate the camera about the rail’s pivot and watch whether they stay lined up.

If the pivot is at the entrance pupil, they do, at every rotation, exactly. If it is not, the near one slides against the far one, and the amount it slides is a parallax: proportional to the pivot’s offset and inversely proportional to the near object’s distance. Slide the camera along the rail until the sliding stops; the pivot is now at the pupil, and the rail’s scale reads off where that is relative to whatever the camera is mounted by.

The reason this works is the same reason the dome works, run backwards. A rotation about the pupil leaves every ray of the picture in the same place relative to the camera — because a ray of the picture passes through the pupil, and rotating about a point on a line does not move the line. A rotation about any other point translates the pupil sideways, and a translated centre of projection is a different view of a three-dimensional scene, which is what parallax means.

The lens field measures that in pixels rather than by eye: 50 mm off the pupil, 12° of rotation, and the nearest post lands 3.2 px from where the other frame put it, with the error falling exactly as one over the distance.

The pattern: a surface that does nothing #

Stand back and there is a shape worth extracting from this, because it recurs.

Every optical element in this field costs something, and what it costs is a function of the angle of incidence. A flat interface presents every ray with a different angle of incidence, so it does something different to each of them, and the varying effect across the field is what the word distortion names.

A spherical interface centred on the projection’s own centre presents every ray with the same angle of incidence, and that angle is zero. So the effect is the same for every ray, and the same as nothing.

Read that way, the dome port is not a clever correction; it is the observation that a projection has a centre and that a sphere about a point is the surface every ray through the point crosses normally. The optics is one line of Snell’s law; the geometry is the whole of it.

The same idea appears elsewhere on this site with the roles exchanged. An anamorph in a cylindrical mirror is the opposite trick — a surface chosen so that its effect on each ray is maximally different, so that a scrambled drawing unscrambles from one place and nowhere else. Both are the same observation: the effect of a surface is decided by the relationship between its normals and the projection’s centre.

The field, ending #

This field opened by breaking the site’s premise and it closes with a case where the premise is restored exactly.

In between: a picture through water whose rays miss their own centre by ten millimetres; a pane of glass that preserves every direction and no position, splitting the site’s two recoveries cleanly in half; a sky compressed into a cone with an area scale that runs to zero; a flat window that costs a third of a lens’s field and cannot be corrected by any warp of the image.

The instrument that produced all five is one function — a stack of parallel layers and a bisection — plus one three-line vector form of Snell’s law. That the same code covers a swimming pool, a display case, a diver’s mask and a camera housing is the claim these essays are making as much as any of the individual numbers: they are not four phenomena, they are one, and the differences between them are entirely in the list of layers.

The next field takes the other departure from the pinhole, the one that is present in every photograph anybody has ever taken rather than only in the wet ones.