A straight stick in water is a kink and a curve

The oldest picture of refraction is a stick standing in water and appearing broken at the surface. What a ray does at a surface went one step past the usual account of it and said, without measuring, that the submerged part does not appear straight-but-tilted: every point of it is seen along a slightly different line of sight, so every point is displaced by a different amount, and the submerged part should appear curved. It called this the first place the field’s central claim shows up without equipment — a straight thing in water is not imaged as a straight thing.

That essay stated the curve. This one traces it, point by point, and finds that “bent” is two different things with two different rules. There is a kink at the surface, with a closed form that does not care which way the stick leans. There is a curve below the surface that cares a great deal. And there is a photograph, which sees one of them plainly, the other barely, and — for one family of sticks — neither at all.

Every point has its own image #

A point under water has two depths found that a single submerged point seen along a line of sight has two images: a sagittal one, on the point’s own vertical, which two level eyes triangulate; and a tangential one, shallower and nearer the viewer, which a nodding head triangulates. A stick is a line of submerged points, each seen along its own line of sight, so it has a line of sagittal images and a line of tangential ones. Those two loci are the two sticks a viewer can see.

The figure traces them. For each of 64 points along the submerged part, it finds the ray that reaches the eye through the site’s one layered solver, places the point’s sagittal image both on the point’s own vertical at the closed-form depth and along the traced ray at the closed-form distance, and checks that those two placements are one point. They are, to a nanometre. Then it measures three things about each locus: the kink, the angle between the stick and its image where the image leaves the surface; the turn, how far the image’s own direction rotates between the surface and the tip, which is zero for any straight line however it is tilted; and the sag, the image’s largest distance from its own chord.

That the image is a line of separately placed points, rather than the stick moved as a whole, is the whole difference from glass. What survives a pane of glass found that a slab moves every point and keeps every direction, because light leaves the far face parallel to the way it met the near one and the bend at the first surface is undone at the second. A viewer outside water looking in has no second surface. The bend at the surface is never undone, a picture through water has no viewpoint found that no single centre explains what arrives, and so nothing obliges a straight row of points, each placed along its own line of sight by its own amount, to stay a straight row.

For the stick drawn — 30° from the vertical, leaning across the line of sight — the sagittal image kinks 14.96° at the surface, turns 4.29° more along its length and sags 6.8 mm from its chord. Its tip is seen 0.549 m down; the stick’s tip is 0.866 m down. The image is shorter, shallower, bent at the surface, and bent again below it.

The kink has a closed form with no direction in it #

Near the surface, every point of the stick is seen along very nearly the same line of sight as the point where it enters the water. The sagittal depth of a point along a fixed line of sight is its true depth multiplied by one factor,

$$k = \frac{\cos\theta_{S}}{\sqrt{n^{2} - \sin^{2}\theta_{S}}},$$

where $\theta_{S}$ is the line of sight to the entry point — the same factor $\cos\theta/(n\cos\theta_{w})$ the two-depths essay derived, written without the water angle. So just below the surface the image is the stick with its depths squeezed by $k$ and its horizontal positions untouched. A line leaning at $\beta$ from the vertical, squeezed vertically by $k$, leans at

$$\tan\beta' = \frac{\tan\beta}{k},$$

and the kink is $\beta' - \beta$. Nothing in that expression says which horizontal direction the stick leans in. A vertical squeeze tilts every line by an amount set by how far it leans, not by which way.

For this stick $\theta_{S}$ is 51.34°, $k$ is 0.578, and the stick leaning 30° appears to leave the surface leaning 44.96°: a kink of 14.96°.

The traced kink agrees with the closed form at every direction swept, and across the whole sweep from leaning straight away to leaning straight toward the eye it varies by less than a millionth of a degree. That is a genuine test rather than a restatement: the trace never uses the formula, and a mistake in the surface-crossing geometry would make the kink depend on the lean’s direction at once. A first version of the measurement did exactly that, by reading the kink over the first of 80 samples rather than at the surface, and found a spurious 0.08° of dependence; the tangent is now taken over a ten-millionth of the stick’s length.

The curve below the surface does care #

Further down the stick the approximation that made the kink simple fails. A point halfway down is not seen along the entry point’s line of sight; it is further from or nearer to the eye, seen more or less obliquely, and its depth is squeezed by a different factor. The squeeze changes along the stick, so the image keeps turning.

How much it turns depends on the direction of the lean, because the direction decides how fast the line of sight changes along the stick. Leaning straight away from the eye, the image turns 1.11° between the surface and the tip and sags 1.7 mm. Leaning across the line of sight, 4.29° and 6.8 mm. Leaning straight toward the eye, 7.62° and 12.7 mm, and its tip is seen 0.592 m down.

So the everyday description has the parts the wrong way round. The kink — the part everyone describes — is the simple, direction-free part with a formula. The curve — the part the earlier essay stated and nobody draws — carries all the dependence on the stick’s orientation.

The stick a nodding head sees #

The tangential locus is a different stick again, and a far less tidy one. Its kink at the surface is 9.6° when the stick leans away from the eye, 36.9° across the line of sight and 38.5° toward it, and its sag is 1.2 mm, 31.4 mm and 58.7 mm. The tangential image of a point lies off the point’s own vertical, displaced toward the viewer by an amount that grows with obliquity, so a line of them is pulled sideways as well as squeezed, and the pull depends on direction from the start.

This is the practical content of the two-depths result applied to an object with extent. Two level eyes and a nodding head do not merely disagree about how deep the stick goes; they disagree about its shape, its angle at the surface and how bent it is, and only one of the two has a kink that ignores the lean’s direction. If a pair of eyes tilts, its two rays to each point of the stick miss each other, and the stick it triangulates is a compromise between two shapes that no point of the water holds.

A peak, and a turn that vanishes while the image is still bent #

Swept through its lean rather than round its direction, the stick shows two behaviours worth measuring rather than assuming.

The kink is not monotone in the lean. An upright stick has no kink, because a vertical line squeezed vertically is still vertical; a stick lying almost flat has almost none, because a nearly horizontal line squeezed vertically stays nearly horizontal. In between the kink rises to a peak of 15.50° at a lean of 37° and falls to 2.10° at 85°. A stick at 30° and one at 45° kink by the same 14.96°, which is a small coincidence of this peaked curve and a warning against reading a kink back into a lean.

The turn is stranger. It peaks near a lean of 25° and falls, and near 66° it passes almost through zero — 0.004° between the surface and the tip. A test that asked only whether the image ends pointing the way it started would call that image straight. It is not: it still sags 0.43 mm from its chord, curving one way near the surface and back the other way near the tip, so that its end directions happen to agree.

That is the reason the figures report sag as well as turn, and it is the general lesson of every straightness test. Fitting a lens from straightness alone measures a line’s straightness by the distance of its points from a line, not by comparing its ends, because an S-shaped departure passes an end-to-end test exactly.

What a photograph from the eye shows #

Everything so far is about images a pair of eyes triangulates. A photograph is something else: a record of the directions in which the stick’s light arrives at one point. Directions are the same for both focal images — both lie on each point’s chief ray — so a photograph from the eye shows neither the sagittal stick nor the tangential stick. It shows the pencil of chief rays, cut by a picture plane.

In that photograph the stick leaning across the line of sight is plainly broken: the submerged part leaves the line of the dry part by 15.44° and its tip ends 37.0 px off that line. The curve is almost invisible. Along the 166 px of submerged stick the image turns 4.50°, but it sags only 1.63 px from its own chord — a departure from straightness under two pixels, at a scale where a thin stick’s own width and blur are larger. A photograph of a stick in a pool is a photograph of a kink.

Leaning 45° round from straight away, the photograph’s kink is larger, 26.41°, because the submerged part is foreshortened into 93 px and a small displacement is a large angle on a short segment; its sag is 0.82 px.

The stick the photograph cannot bend at all #

One family of sticks photographs as perfectly straight, and the reason is worth stating before the figure, because it is a consequence of the geometry rather than a finding.

When the stick leans straight toward or straight away from the eye, the stick and the eye lie in one vertical plane. Every ray from every point of the stick to the eye stays in that plane — it is the plane of incidence for each of them, and refraction never takes a ray out of its plane of incidence — so every chief ray lies in one plane through the eye. A plane through a centre of projection is photographed as a line. The dry part, the entry point and the whole submerged part are therefore photographed on one straight line, whatever the water does to them.

The figure’s measurement confirms that construction to $7 \times 10^{-14}$ px for the stick leaning away and $10^{-13}$ px for the stick leaning toward the eye. That is a check on the photograph code rather than evidence about water: it would hold for any refraction that keeps rays in their plane of incidence. What it says about sticks is the useful part. A pole leaning straight toward or away from a camera looks unbroken in the photograph, and it is exactly that pole whose sagittal image — the one two eyes side by side would triangulate — is kinked by the full 14.96°. One eye sees nothing wrong with it; two eyes see it bent.

This is the same structure straight lines that are not found for a lens: a radial distortion bends every straight line except those through the principal point, because those already lie along the direction of the map. Here the privileged lines are the ones whose plane contains the eye, because those already lie along the plane the refraction works in.

A kink with no curve, and a curve with no corner #

It helps to set the stick beside two other straight things that come out bent, because the comparison says what kind of map refraction is.

A shadow across an edge found that a straight rod’s shadow falling across the crease between a floor and a wall is two straight pieces, each dead straight, meeting at an angle. That is a kink with no curve at all, and it has to be: a shadow is a projection from a point onto a surface, a projection sends lines to lines, and two flat surfaces give two straight pieces. A wire with a corner in its shadow found the converse — a smooth wire whose shadow acquires a corner where its tangent runs along the light’s ray.

The stick has both a kink and a curve, and the curve is the evidence that refraction is not a projection from any point. The kink alone could be imitated by a projection onto two planes. The curve cannot: no centre and no arrangement of flat picture surfaces sends a straight line to a line that keeps turning. That is the earlier essay’s claim, now with its size attached — and the size, in a photograph, is under two pixels, which is why the stick has been described for so long as merely broken.

The control, and where the numbers move #

Every one of these measurements runs through the same code with the water replaced by air.

With an index of 1 the traced image of the stick is the stick, its kink, turn and sag all at the arithmetic floor. That is the check that the kink and curve above are the water’s and not the computation’s — the same shape as the control every refraction figure here draws beside its measurement, and the same shape as one surface, two images, where the reflected half of a water photograph has a centre and the refracted half does not.

The kink grows with distance from the eye, because a more distant entry point is seen more obliquely and squeezed harder. Entering the water 1 m from the eye, where the line of sight is 32.0° from the vertical, the stick kinks 9.79°; entering 4 m away, at 68.2°, it kinks 26.08°. In a liquid of index 1.52 the stick at 2 m kinks 20.32°. The curve’s turn changes much less: 4.29° in water and 4.37° at the higher index.

What this model leaves out #

The stick is a line of points; a real stick has thickness, and its two edges are two such lines seen at slightly different angles. The surface is flat and still; a ripple of a millimetre changes the line of sight to each point by more than any of the turns measured here. The two images are the limits of a thin pencil, which two eyes 65 mm apart at two metres reach closely and a wide camera baseline would not. And nothing here says what a human visual system does when two eyes are handed the sagittal stick and one eye the photograph’s straight line; that is a question about perception, and the collection’s standing limit applies to it.

Still open: what stick a stereo camera reconstructs #

This essay found two sticks, one for each focal line, and a photograph that shows neither. A stereo camera, whose baseline need not be level, reconstructs a third. For each point of the stick its two rays miss each other unless the baseline is exactly level or exactly upright, and the reconstruction takes the midpoint of the miss — which the midpoint is a choice of ruler found to be a convention rather than a measurement. The open measurement reconstructs the stick from a stereo pair whose baseline tilts from level to upright, measures the reconstructed kink, length and sag at each tilt, and finds the tilt past which the rays’ miss along the stick exceeds the disparity a pixel represents at that range — the point at which the reconstruction is no longer a stick that a better matcher could sharpen, but a shape no point in the water supports.