Two pictures on one screen

A stereoscopic display shows each eye a different picture. Everything else about it — the polarisers, the shutters, the lenticular sheets, the headsets — is machinery for arranging that one fact, and none of it changes the geometry.

The geometry is a triangle. The two eyes are at known places; each is shown a mark at a known place on the screen; the depicted point is where the two sightlines cross. One similar-triangle argument gives the whole thing:

$$z = \frac{b\,D}{b - d}$$

with $b$ the separation of the eyes, $D$ the distance to the screen and $d$ the horizontal separation between the two images of the point on the glass.

Three regimes, all visible in one formula #

$d = 0$ puts the point at $z = D$ — on the screen. So the screen plane is the zero-disparity plane, and it is the one depth a stereoscopic display renders with no burden on the viewer at all. Everything a designer wants to be comfortable goes there.

$d < 0$ — the left eye’s image to the right of the right eye’s, which is crossed disparity — brings the point in front of the screen. The formula handles it with no special case: a negative $d$ makes the denominator larger than $b$ and $z$ smaller than $D$.

$d \to b$ sends the denominator to zero and the point to infinity. And $d > b$ makes the denominator negative, which the formula reports as a point behind the viewer and which the eyes report as an instruction to point outward.

That third regime is the one worth dwelling on.

The ceiling is a head #

Eyes converge inward easily. They do not diverge — the resting state is very nearly parallel and outward rotation past parallel is not available. So a disparity greater than the interocular distance is a request the visual system cannot fulfil, and what it produces is not a very distant object but a failure to fuse.

The formula says the boundary is at $d = b$ exactly, and it says so with $D$ absent. The gate asserts that: the same 63 mm disparity reaches infinity at a screen 2 m away and at one 6 m away.

That is a genuinely odd and rather good result. A stereoscopic display’s maximum usable disparity is a property of the reader’s skull, and does not change when the screen moves, or grows, or changes resolution. Every other quantity in display engineering scales with something; this one is 63 mm, on any display, for ever.

Its practical form is the one cinema learned expensively. A film graded for a cinema screen has disparities measured in fractions of the screen width; shown on a television a tenth as wide, those fractions become a tenth as many millimetres and everything moves toward the screen plane. Shown on a screen wider than the one it was graded for, they become more millimetres — and the ones near the ceiling go past it. A stereoscopic film is correct for one screen width and cannot be correct for two, which is a much stronger statement than the one this site makes about ordinary photographs, where the wrong width merely stretches depth.

This is the site’s own stereo geometry, read backwards #

The depth field already built this. Depth is a reciprocal takes a stereo camera pair with baseline $b$ and focal length $f$, and derives

$$z = \frac{b\,f}{d}$$

which is the same triangle with the eyes replaced by cameras and the screen by an image plane.

A stereoscopic display is that construction with the arrows reversed. The camera pair converts a depth into a disparity; the display converts a disparity back into a depth; and the two are exact inverses when the display’s geometry matches the camera pair’s. Match means: the ratio of screen distance to screen width equals the ratio of the cameras’ focal length to their sensor width, and the eye separation equals the camera baseline.

Almost nothing meets both conditions. A stereo pair shot with a 65 mm baseline — human separation, the obvious choice — and shown on a screen the viewer sits closer to than the shot’s station point produces a scene compressed in depth; shown further back, a scene stretched. And a pair shot with a wider baseline for a distant subject produces, on any display, a scene depicted at a smaller scale than life, which is the “miniature” look of long-baseline stereo. Every one of those is the same mismatch this field has measured twice already, with a second projection in place of the first.

Depth resolution, from the same curve #

The curve is a hyperbola, so its slope is not constant, and the slope is what says how much depth a given change in disparity buys.

$$\frac{\mathrm{d}z}{\mathrm{d}d} = \frac{b\,D}{(b-d)^2} = \frac{z^2}{b\,D}$$

Quadratic in the depicted depth — the same shape as a stereo pair’s depth uncertainty, and for the same reason, since it is the same triangle.

Which has a consequence for a display that the camera-side essay states for a measurement. A display with a finite pixel pitch has a finite disparity resolution, so it has a depth resolution that degrades as the square of the depicted distance. On a screen 2 m away with 0.25 mm pixels, one pixel of disparity distinguishes depths of about 8 mm at the screen plane and about 2 m at a depicted 30 m.

So a stereoscopic display renders depth finely near the glass and coarsely far behind it, and the transition is quadratic. That is the same lopsidedness a depth buffer has, arrived at from a completely different direction — one is a quantisation of $1/z$ in a register and the other is a quantisation of disparity on a panel, and both come out as $z^2$.

The reason they agree is not a coincidence and is worth stating: both are quantisations of something proportional to $1/z$, and differentiating $1/z$ gives $-1/z^2$ whatever the something is.

The other distance, and the split between them #

There is a second geometric fact about a stereoscopic display and it is the one everything else about the technology is organised around.

The eyes focus on the screen and converge on the depicted point. Those are two different distances the moment the disparity is not zero, and in ordinary vision they are always the same distance — the mechanism that focuses and the mechanism that converges are yoked, because in a world made of real objects the thing being looked at is at one distance.

The gap is conventionally measured in dioptres, which is reciprocal metres, because that is the unit accommodation works in. With the screen at 2 m the eyes accommodate at 0.50 D whatever is depicted, and converge at $1000/z$ mm⁻¹.

The last row is worth reading twice. At a screen 2 m away, the split for something depicted at infinity is exactly 0.50 D — which is $1000/2000$, the screen’s own dioptric distance. That is not a coincidence and it is not an approximation: anything depicted behind the screen has a split bounded by the screen’s own dioptric distance, because vergence for a distant object goes to zero and the focus stays where the glass is.

So a display 2 m away has half a dioptre of headroom for the whole of the space behind it, and a display 400 mm away — a desk monitor — has two and a half dioptres. Which is why near stereoscopic displays are harder than far ones, in a way that has nothing to do with the display.

The split is linear, and it decides which limit binds #

The table above is four rows of a straight line, and the line is worth having because it turns the comfort rule into arithmetic.

Focus is fixed at the glass, $1000/D$ dioptres. Vergence is $1000/z$ with $z = bD/(b-d)$, so $1000/z = 1000(b-d)/bD$, and the difference is

$$\text{split} \;=\; \frac{1000\,|d|}{b\,D}\ \text{dioptres},$$

with $d$ and $D$ in millimetres. Every $D$ and every $b$ has dropped out except as that product: the split is proportional to the disparity and to nothing else about the depicted point. Against the table, 20 mm at a 2 m screen gives 0.159 D and 30 mm gives 0.238, where the measurement reports 0.16 and 0.24. It is the same line on both sides of the glass, which is why the last column is symmetric in a way the depth column is violently not — depth is a reciprocal and the split is not.

Now set it against the other limit. A stereoscopic display has two independent ceilings on disparity: the skull, at $d = b$, beyond which the eyes are asked to diverge; and comfort, at whatever split $S$ the reader tolerates, which the industry puts between a third and a half of a dioptre. The second gives

$$|d|_{\max} \;=\; \frac{S\,b\,D}{1000},$$

and the two ceilings are equal when $SbD/1000 = b$ — that is, at

$$D \;=\; \frac{1000}{S}\ \text{mm} \;=\; \frac{1}{S}\ \text{metres},$$

the dioptric distance of the comfort limit itself, with the eye separation cancelled out of both sides. At half a dioptre that is two metres, which is why the 63 mm row of the table and the 0.50 D row are the same row and why that looked like a coincidence.

It is the boundary between two regimes, and which side of it a display sits on decides everything about how it must be authored.

Nearer than 1/S — every desk and every headset. Comfort binds long before the skull does. A monitor at 400 mm allows 12.6 mm of disparity, a fifth of the divergence budget, and the depicted window that fits inside it runs from 333 mm to 500 mm: six centimetres in front of the glass and ten behind it, on a screen the reader has their hands beside. The whole depicted world has to live in that slab, which is why near stereo is authored so conservatively and why it is not a failure of craft.

Further than 1/S — a cinema. The skull binds and comfort never does. At fourteen metres, 63 mm of disparity costs 0.07 D of split, so everything from about 1.75 m in front of the audience out to infinity is inside both limits at once. The window is the whole scene.

The ratio of the two ends makes the same point in one expression. In the comfort-bound regime the depicted range runs from $bD/(b + |d|_{\max})$ to $bD/(b - |d|_{\max})$, a ratio of

$$\frac{1 + SD/1000}{1 - SD/1000},$$

which is 1.5 at 400 mm, 3.0 at a metre, and unbounded exactly at two metres. A stereoscopic display’s usable depth is not proportional to its viewing distance; it diverges at a finite one. That is the strongest form of the sentence the chain from sensor to screen makes about ordinary pictures — a display’s geometry is decided by where the reader sits — and here it decides not a factor but which of two unrelated constraints is the one being designed against.

Why the screen plane is a design decision #

One consequence of the arithmetic is worth stating as a rule, because it is the one thing a designer of a stereoscopic picture actually controls.

The zero-disparity plane is where the depicted scene meets the glass, and moving it costs nothing geometrically — shifting both images horizontally by equal and opposite amounts moves the whole depicted scene toward or away from the viewer, uniformly in disparity, without changing any shape.

So the whole depicted scene can be slid along the depth axis freely, and the constraint is at the ends: nothing may exceed 63 mm of disparity behind, and nothing may come so far forward that the vergence split becomes unreasonable. The design decision is where inside that window to put the scene, and it is a real choice with no geometric preference.

The convention that emerged in cinema is to put the screen plane near the front of the depicted scene, so that most of the action is behind the glass. That places the burden on the divergence limit rather than on the crossed-disparity one, which is the more forgiving of the two — the window behind the screen is bounded by the head’s width and the one in front is not bounded at all, only made progressively more demanding.

Where this site stops #

The comfort question is not a geometric one and this site does not answer it.

What is computed here is a split between two distances, exactly, in a stated unit. Whether a viewer finds a given split comfortable, how quickly they adapt, whether it causes fatigue, and how it varies between people are questions about the visual system, and the site has said since its foundation that it computes the geometry of pictures and says nothing about how one is perceived.

The half-dioptre figure quoted above is the industry’s rule of thumb and is quoted as such — it is where a line is conventionally drawn, and this site has no standing to say whether it is drawn in the right place. What the geometry supplies is the ability to say where any particular disparity falls relative to whatever line somebody else draws, which is the useful half.

That boundary is the same one the site draws everywhere. Wide-angle is not distortion computes the stretch and declines to say whether it looks bad. Standing in the wrong place computes the depth factor and notes that viewers mostly do not notice. Geometry produces numbers; what to do about them is somebody else’s field.

The two projections a stereoscopic display has #

Everything in the screen field so far concerns a picture projected twice — once by the camera and once off the glass. A stereoscopic display has the second projection twice over, and the two copies do not have the same station point.

Each eye sees the screen from its own position, so each has its own station distance and its own angle to the glass. For a viewer square to a screen at 2 m with eyes 63 mm apart, the two eyes are 31.5 mm either side of the centre line — a difference in station point of 63 mm on a distance of 2000, which is 1.8°.

That is small and it is not nothing, and it has a consequence with a name. The two pictures must be rendered from two different virtual cameras, and those cameras must be parallel with offset principal points rather than converged. Toe the two cameras inward to meet at the subject and the picture planes are no longer coplanar, so the two images of a point differ vertically as well as horizontally — and a vertical disparity is one the eyes cannot fuse at all, at any size.

The geometry says so directly: two cameras with coplanar image planes give correspondences differing only along the baseline, which is the epipolar geometry of a rectified pair. Toe them in and the epipolar lines are no longer horizontal, and every point off the convergence plane acquires a vertical component.

So the standard advice — never converge the cameras, shift the frusta instead — is not a preference. It is the requirement that the pair be rectified, and rectification is exactly the condition that the two pictures can be fused by two eyes that cannot rotate about the line between them.

What two pictures buy, and what they still do not #

Ending on the thing this field has in common with the twoviews field, because it is the same limit seen from the display’s side.

Two pictures of a scene determine its shape and not its size — that is the two-view field’s central result, and it is a statement about what a pair of pencils of rays contains.

A stereoscopic display has the same property from the other end. Two pictures shown to two eyes determine a shape; what fixes the size is the eye separation, which the display does not control and cannot know. A viewer with eyes 58 mm apart and one with eyes 70 mm apart, watching the same film from the same seat, are shown two scenes of different sizes — the same shape, scaled by the ratio of their separations, which is about 20%.

That is a genuinely peculiar property of the medium and it falls straight out of the formula: $b$ appears in the numerator, so $z$ scales with it. Nothing else in the history of pictures has the property that the depicted object’s size depends on the width of the observer’s head.