A point under water has two depths

What a ray does at a surface replaced the textbook’s apparent depth with a curve. Dividing by the refractive index is right only looking straight down; along a line of sight $\theta$ from the vertical, a point $h$ down appears at

$$d' = h\,\frac{\tan\theta_{w}}{\tan\theta},$$

which for a point 1.50 m down in water is 1.125 m straight down, 0.740 m at 60° and a fifth of the true depth at 80°. The bottom of a pool is not a plane at $h/n$, that essay concluded, but a curved surface rising toward the far end.

That curve still gives each point one apparent position. It is found by taking the ray that reaches the eye, extending it back into the water, and asking where it reaches the point’s own vertical. One ray has one extension, and one extension meets one vertical once. But an eye does not see a point along one ray. It sees it through a pupil, and depth is read across two positions — two eyes, or one eye that moves. The question this essay puts to the curve is whether the rays that reach a small pupil, extended back into the water, meet at the position the curve gives. They do not meet at any single position at all.

A pencil with two waists #

Take the thin bundle of rays a submerged point sends to a small pupil. Each ray leaves the point in a slightly different direction, reaches the surface at a slightly different place, and bends there by Snell’s law. Extended back into the water, the rays would meet at one point only if the surface imaged the point without aberration, and a flat surface seen obliquely does not.

The bundle has two planes worth separating. Rays fanned up and down within the vertical plane containing the line of sight — the tangential fan — meet the surface at different angles of incidence, and the bend grows steeply with incidence. Rays fanned sideways, out of that plane — the sagittal fan — all meet the surface at nearly the same incidence, and are turned only because the surface’s normal is tilted relative to each of them. The two fans are bent by different amounts, so their extensions converge at different distances. Each fan comes to a focus along a short line, and the two focal lines lie at different depths.

For a flat surface both positions have closed forms — the flat-surface case of the equations optics uses for a tilted pencil through any refracting surface:

$$d_{s} = \frac{h\cos\theta}{n\cos\theta_{w}}, \qquad d_{t} = \frac{h\cos^{3}\theta}{n\cos^{3}\theta_{w}}.$$

Looking straight down both are $h/n$, and they agree. Off the vertical the second always falls faster than the first, because it carries the cube.

At 30° from the vertical the sagittal image of a point 1.50 m down is 1.051 m down and the tangential 0.918 m. At 60° they are 0.740 m and 0.320 m — the second less than half the first. At 80° they are 0.290 m and 0.019 m, the tangential image lying almost on the surface.

The curve that was already drawn is the sagittal one #

Set the two formulas beside the earlier apparent-depth curve. With $\sin\theta = n\sin\theta_{w}$,

$$\frac{\tan\theta_{w}}{\tan\theta} = \frac{\sin\theta_{w}\cos\theta}{\cos\theta_{w}\sin\theta} = \frac{\cos\theta}{n\cos\theta_{w}},$$

so $d' = d_{s}$ exactly. The apparent-depth curve is the sagittal image, and the construction that produced it — one ray, extended to the point’s own vertical — could only ever find the image that lies on that vertical. The sagittal image does; the tangential image does not.

That identity is also a trap, and it is worth naming because it would be easy to fall into here. Comparing the sagittal formula with that earlier curve and finding agreement to fifteen digits would look like a confirmation. It is the same expression written twice, and it would agree for any refractive index, any angle and any mistake that both inherited. A comparison that can fail has to reach the two depths by a route that shares no arithmetic with the formulas — and the obvious one is to do what a viewer does.

Which of the two a pair of eyes reads #

A viewer reads depth by triangulation: two positions a baseline apart, two lines of sight, and the place where they meet. So the figure places two eye positions, traces the actual ray from the submerged point to each through the same layered ray solver behind every refraction figure here, extends both back into the water, and finds where they meet.

Put the two positions side by side, a millimetre apart on a horizontal line across the line of sight — the way two eyes sit in a level head. Their two rays belong to the sagittal fan, and they meet at the sagittal depth: at 60°, the traced pair reads 0.7401 m against the closed form’s 0.7401 m.

Put the two positions one above the other, a millimetre apart — a head moving up and down. Their rays belong to the tangential fan and meet at the tangential depth: 0.3202 m against 0.3202 m. With a full 65 mm stroke, the traced nod reads 0.3200 m, the difference being the pencil’s width.

Across every angle from 10° to 80° both traced readings land on their closed forms to within two parts in ten thousand of the true depth. Neither route knows about the other. That agreement is what establishes the two formulas, and it establishes something more useful at the same time: which depth a viewer gets is decided by the direction of the baseline across which depth is read, not by the water and not by the point.

A tilted pair’s rays do not meet #

Between side by side and one above the other lies every tilt of a head, and there the two focal lines stop being a curiosity.

Tilt the pair and its two rays are no longer in one fan. One eye’s ray crosses the sagittal focal line at one place and the tangential focal line at another; so does the other eye’s, at different places; and two lines passing through two different pairs of points on two skew focal lines are, in general, skew themselves. The traced rays of a pair tilted 30° miss each other by 5.81 mm. At 40° the miss is 6.39 mm. Upright again, they meet.

What the triangulation returns in between is the midpoint of the shortest segment joining the two rays: 0.601 m at a 30° tilt, 0.489 m at 45°, sliding smoothly from the sagittal depth to the tangential one. The closed form of a thin astigmatic pencil predicts that slide without tracing anything — the depth and the miss both depend only on the two focal distances and the tilt — and the traced curve lies on it.

That midpoint is not a measurement of anything in the water. The midpoint is a choice of ruler found that the midpoint of two rays that miss moves when the frame it is measured in is changed by anything other than a similarity, so it is a convention about how to split an error rather than a fact about a point. Here the convention has nothing to be right about. No point in the water lies on both rays, so there is no depth for a better estimator to recover. A tilted head is not measuring the depth of a submerged point imprecisely; there is no single depth for it to measure.

A baseline has a direction as well as a length #

The stereo essays here have mostly treated a baseline as a length. The range a pair cannot see past fixed the distance beyond which a rig cannot even say “no further than” from its focal length and its baseline, and a turn of the head is not a step sideways measured what a very short baseline does and does not cost a reconstruction. Direction entered through the pictures rather than the depth: what the two eyes are sent found two seats 63 mm apart receiving different maps of one curved screen, and both coordinates agree on a circle and a line found that counting the vertical disparity between two eyes changes where they agree.

In air, turning a baseline within the plane perpendicular to the line of sight changes which rows of two pictures correspond, and nothing about the depth they give: the rays from a point meet whatever the baseline’s orientation, because they all pass through the point. Under water that stops being true. The depth a baseline reads is a function of its direction across the line of sight — the sagittal depth for a level baseline, the tangential depth for an upright one — and only those two orientations have rays that meet at all.

So a reconstruction through a flat water surface has to know not only how far apart its viewpoints were, but which way the displacement between them ran relative to the plane of each look. A rig whose baseline is level for one point and tilted for another, because the points lie in different directions from it, is triangulating two different kinds of image in one scene, and its misses are not noise.

Where the two images sit #

The two images do not only differ in depth. They lie at different places across the pool.

Both focal lines are crossed by the chief ray’s own extension — the backward continuation of the single ray that reaches the middle of the pupil. The sagittal image lies where that extension meets the point’s own vertical, which is why the earlier single-ray construction found it. The tangential image lies on the same extension but much nearer the surface, and therefore nearer the viewer: at 60°, 0.727 m nearer across the pool. The figure places the tangential image along the traced extension at the closed-form distance and checks that it lands at the closed-form depth; the library places the sagittal image both on the vertical and along the ray, and checks that those are one point.

So a viewer nodding at the edge of a pool does not see the point shallower and in the same place. It sees it shallower and displaced toward itself by most of a metre. A viewer walking along the edge, with the point off to one side, sees it at the sagittal depth and on its true vertical. The two images are two different locations, and which one a given motion reveals is fixed by the geometry of that motion.

How far across the pool the second image lies #

The distance between the two images across the pool grows far faster than their difference in depth, and it is the larger effect of the two.

At 30° from the vertical the tangential image lies 0.077 m nearer the viewer than the sagittal one. At 40° it is 0.194 m nearer; at 50°, 0.404 m; at 60°, 0.727 m; at 70°, 1.144 m; and at 80°, 1.535 m — further across the pool than the point is deep. Near grazing the tangential image lies almost on the surface and more than a water’s depth toward the viewer, so the two images of one point sit at the corners of a triangle whose sides are a sizeable fraction of the pool.

That matters for anything read across a pool rather than down into it. A mark on the floor far across the water, seen by a head that bobs as it walks, has a tangential image that swings toward and away from the viewer through most of a metre as the line of sight changes, while its sagittal image — the one a level pair of eyes triangulates — stays on the mark’s own vertical and only rises and falls. Whether that difference is noticeable to anyone looking is a question about perception, and nothing here answers it. What the geometry settles is that the two readings are not a small correction to each other at the angles a far end of a pool is seen from.

At a gentler angle #

The effect is not confined to grazing views.

At 30° from the vertical — a view down into a pool from a metre or so back — the sagittal image is 1.051 m down and the tangential one 0.918 m, a difference of thirteen per cent, and they are 0.077 m apart across. A pair of eyes tilted between the two readings has rays that miss each other by as much as 1.68 mm.

The two images part slowly near the vertical and then quickly. Their ratio is $\cos^{2}\theta/\cos^{2}\theta_{w}$, which is 0.87 at 30°, 0.43 at 60° and 0.07 at 80°. Near-vertical viewing, the case the textbook formula describes, is the case where the distinction is smallest, which is presumably part of why one number served for so long.

A camera that moves reads the depth its motion chooses #

A pair of eyes is one instrument that reads depth across a baseline. A camera that moves is another, and it has no reason to move horizontally.

A camera carried along the edge of a pool, or flown along the bank of a clear river, photographs a submerged point from positions spread horizontally across the line of sight: its baseline is sagittal, and a reconstruction from its pictures places the point at the sagittal depth. The same camera raised and lowered on a mast, or flown toward the water along its line of sight and climbing, has a baseline with a vertical component in the plane of the look, and its reconstruction drifts toward the tangential depth. A reconstruction that combines both motions receives rays that do not meet, with the misses growing as the view becomes more oblique.

That is a prediction rather than a measurement of any particular survey, and it rests on the thin-pencil geometry of a calm flat surface. But it has a practical shape: a depth recovered through water by photographs is not just biased by refraction, it is biased by an amount that depends on the direction the camera moved, and two surveys of the same pool flown differently would disagree about its depth without either being in error. A picture through water has no viewpoint already established that no single centre explains a refracted picture; this is the same failure seen from two viewpoints at once, and it says which way the missing centre moves.

What the two images are, and what they are not #

It is worth being exact about what this changes and what it does not.

It does not change where a point appears to one eye. A single small pupil sees the point along one direction — the direction of the chief ray — and the two focal lines are both on that direction. A photograph from one position, being a record of directions, is identical whichever image is taken to be the real one. The two depths appear only when two positions are compared.

It is the pencil’s geometry, not an effect of the eye. Rays that fail to meet at a point have been met before: where the focus went found that a curved mirror has a caustic instead of a focus, and a ball of water has no eye either found rays through a sphere missing any common point. Those were failures across a wide aperture. This one survives in the thinnest pencil there is, because it comes from the tilt of the surface rather than its extent.

And it is a thin pencil through a calm flat surface. The closed forms are the limit of an infinitesimal pupil; a 65 mm baseline at 1.5 m depth agrees with them to a fifth of a millimetre, and a much wider baseline would not. Waves, a refractive index that varies with colour, and a surface that is not flat each add their own departures, and none is modelled.

Still open: how deep each of a pool’s two floors looks #

The earlier essay turned the apparent bottom of a pool from a plane into a curved surface. This one has found that there are two such surfaces, one for each focal line, and that a viewer’s own motion chooses between them. Mapping both is what remains. For a flat floor 1.5 m down seen from an eye 1.6 m above the edge, the map traces the sagittal and tangential images of every point of the floor out to the far end, measures the slope each apparent floor presents at every distance, and finds the distance along the pool at which the tangential floor is half as deep as the sagittal one — the point past which a nodding head and a walking one disagree about the depth of the water by a factor of two.