A scroll is not a panorama

Two pictures, both much wider than they are tall, both showing more of the world than any single frame could, and both drawing straight world lines as curves. One of them is a projection through a centre and one of them is not, and no amount of looking at the curves says which.

Why the confusion is worth taking seriously #

The two get conflated in three different literatures and it is worth saying where, because the conflation is not a lazy one.

In accounts of Chinese painting, the handscroll is often described as panoramic, and the word is doing double duty: it means very wide, which is true, and it carries the geometric sense, which is false. The wide sense is the older one and the geometric sense arrived with the camera.

In discussions of photography, the two are genuinely both called panoramas. A rotating-lens camera and a strip camera pulled along a track both produce long pictures and both were sold under the same word, and the second one is a pushbroom.

And in computer vision, the two are cleanly separated — a rotating camera is a homography and a translating one is not — but the separation is stated in a vocabulary that does not reach the other two literatures.

So there is a real distinction, it is exactly stateable, and the statement lives in a place where nobody discussing either object is likely to encounter it. That is the gap this essay is in.

The two objects #

A cylindrical panorama is a camera that turns on the spot. The eye stays where it is; the direction it looks sweeps through an angle; the picture is the record of what arrives at that one point from each direction, laid out on a cylinder and then unrolled. This site already built one, and the essential fact about it is that the eye never moves.

A handscroll is a camera that travels. The direction it looks stays where it is; the eye slides along a track; the picture is the record of what arrives at each successive point from one fixed direction.

Those are exact opposites in a precise sense: one holds the position and varies the direction, the other holds the direction and varies the position. Both produce a long picture. Only the first has all its rays passing through one point, because in the first there is only one point for them to pass through.

Why both curve their lines, for different reasons #

The curvature is where the confusion starts, so it is worth saying exactly what causes it in each case.

In a panorama, the surface is curved. The rays are a perfectly ordinary bundle through a point; what happens to them is that they land on a cylinder instead of a plane. A straight world line and the eye define a plane, that plane cuts the cylinder in an ellipse, and the unrolled ellipse is a sinusoid. Nothing is wrong with the projection at all — the picture is a projection from one point onto a surface that is not flat, and the curved field measures exactly what that costs.

In a scroll, the surface is flat and the eye moves. Each column is drawn from a different point, so there is no single bundle. The image of a straight line is a hyperbola rather than a sinusoid, which is a small difference on the page and a total one in kind.

So the shared symptom has two unrelated causes, and the test that separates them is not about the lines at all. It is about the rays.

The measurement, twice #

assertOnlyOneOfThemHasACentre builds both bundles and hands them to the same solver.

For the panorama, twenty-five bearings across 120° and five elevations across 40°, all from one point. The fit returns that point, and the rms miss is $1.9 \times 10^{-15}$ m — which is the noise floor of a 3 × 3 elimination in double precision and is the number that says yes, exactly.

For the scroll, twenty-five columns across the width of the paper and five rows down it. The fit returns a point nothing passes through, and the miss is 7.97 m.

A difference of fifteen orders of magnitude is not a comparison that needs interpreting. What needs saying instead is that the panorama’s answer is the interesting one: it says that turning a camera as far as it will go — 120°, 360°, any number of full revolutions — costs the picture nothing in this respect. The station point is exactly as much a station point at the end of the sweep as at the beginning.

That is what makes a panorama stitchable, and it is worth spelling out because it is the practical form of the same fact. Two frames from a rotating camera are related by a homography, so they can be aligned without knowing anything about the scene. Two frames from a translating camera are related by a homography only if the scene is planar or infinitely far away, and otherwise the alignment depends on depth — which is the parallax the lens field measures and the reason panoramic heads have a rotation point that has to be found.

The property that is actually shared #

Having separated them, it is worth being fair about what the two do have in common, because it is not nothing and it is what makes the conflation tempting.

Both are unbounded in their long direction. A panorama can be extended to a full turn and beyond; a scroll can be extended for ever. A flat perspective picture cannot be extended past 180° at all, and long before that it runs away — the picture width goes as the tangent of the half-angle, so a 170° flat picture is eleven times wider than a 90° one and the edges are unusable.

Both give up the flat picture plane’s guarantee of straight lines, which is the visible symptom.

And both are read a piece at a time. A panorama on a screen is panned; a scroll is unrolled. Neither is an object anybody looks at all at once, which is a fact about how they are used rather than about their geometry, and it is probably the real source of the conflation.

What they do not share is the one property this site is built on. A panorama has a point from which it is a correct projection of the scene; a scroll does not. That is why the panorama’s figures on this site print a viewing distance and the scroll’s print a miss, and why the gate that polices which figures may print which now demonstrates both.

The four cameras this makes, and the two nobody built #

Setting the two side by side suggests the obvious classification, and it is worth writing out because one cell of it is empty for an interesting reason.

An eye can hold its position or move it, and hold its direction or turn it. Four combinations:

Fixed position, fixed direction is an ordinary camera. One picture, one centre, a bounded field.

Fixed position, turning direction is a panorama. One centre, unbounded field, curved lines, and a picture that is a projection.

Moving position, fixed direction is a scroll, and a satellite. No centre, unbounded extent, curved lines, exact measure along the track.

Moving position, turning direction is a camera being carried through a scene, which is not a picture at all — it is a sequence, and turning it into one picture is the structure-from-motion problem, which needs many frames and recovers the track along with the scene rather than laying anything out on a page.

The fourth cell is empty as a drawing system precisely because it has too much freedom: a system needs a rule, and “the eye goes wherever it likes” is not one. The three that are full are the three with a rule, and their rules are the three ways to constrain an eye to a one-parameter family.

What each of the three does with a wide subject #

The three systems all exist because a flat perspective picture cannot hold a wide scene, so it is worth comparing what each of them does with the width rather than merely how they fail.

A panorama spends angle. Its horizontal coordinate is a bearing, so equal angles get equal paper wherever they are. That is the property that makes it the right instrument for a view: a distant mountain range and a nearby wall occupy paper in proportion to how much of the visual field they take, which is what a viewer standing in one spot experiences.

A scroll spends distance. Its horizontal coordinate is a position along a track, so equal metres get equal paper. A distant mountain occupies almost none, and a nearby wall running along the track occupies its full length. That is the right instrument for a journey, where what matters is how far things are apart rather than how large they loom.

A flat picture spends tangent. Its horizontal coordinate is the tangent of the bearing, which is why the width runs away: equal angles near the edge get vastly more paper than equal angles near the centre.

Three coordinates — angle, distance, tangent — and every property in this essay follows from which one a system chose. A panorama’s straight lines curve because a bearing coordinate is not linear in position; a scroll’s curve because its columns are drawn from different points; a flat picture’s do not curve and its edges stretch instead.

That framing also settles the question of which is “correct”, by dissolving it. Each coordinate is the right one for a different question about the scene, and none of them is a coordinate the world has.

Which of the three a picture is, from the picture alone #

A last question, and it is the one a reader looking at an unfamiliar long picture would actually ask: can the three be told apart without being told which is which?

Yes, and the test is the curvature.

A flat perspective picture draws every straight world line straight. Any curve in the drawn image of a known-straight edge rules it out.

A panorama draws a straight line as a sinusoid on the unrolled surface: the curve is symmetric about its extremum, its amplitude is set by how far the line passes from the eye’s height, and — the diagnostic — a line at the eye’s own height is drawn straight, whatever its bearing.

A scroll draws a straight line as a hyperbola: the curve is asymmetric, with a horizontal asymptote and a vertical one. It draws two families straight rather than one — the lines at constant depth, whatever their height, and the lines at the eye’s own height, whatever their depth.

So the two families are separable from a single well-chosen line, and choosing it takes some care, because the obvious choice does not work. The next section is about which line does.

The line that tells them apart, and the one that does not #

The obvious test is a long straight edge running at eye level away from the viewer, and it is worth working out why it fails before saying what replaces it, because the reason is the same algebra in both systems arriving at the same answer from two directions.

A panorama draws it straight because its across-picture coordinate is an elevation, and a line at the eye’s own height has an elevation of zero at every bearing. The image is the horizon row.

A scroll draws it straight too, and this is the part the obvious test overlooks. Its across-scroll coordinate is $v_{0} + f(h_{e} - Y)/z$, and at $Y = h_{e}$ the numerator vanishes — so every point of the line images to $v_{0}$ exactly, at every depth. Checked against the model on a line receding from 2 m to 20 m: 150.000000 px at all seven samples. Not nearly straight. The same row, to six decimals, whatever the depth does.

So an eye-level edge is drawn straight by both, and by a flat perspective picture as well. It is the one line in the world that every system in this field agrees about, and a test built on it separates nothing.

The line that does separate them is a horizontal edge above or below eye level, running broadside — at constant depth from the track. A scroll draws it perfectly straight, because the depth in its denominator is constant and the whole line images to one row: 71.166667 px at every sample, for a line 3.2 m up at a constant nine metres. A panorama cannot, because its own distance to a straight line varies along it — nearest at the perpendicular foot and further at both ends — so the elevation it records rises and falls, and the drawn edge is a curve symmetric about the nearest point.

That is the diagnostic, and it is the one an actual long picture supplies most readily: the eaves of a hall presented broadside, the top of a wall running beside a road, the line of a parapet. Straight in a scroll; bowed, symmetrically, in a panorama.

Three lines, then, and a reader needs only to know which is which:

At eye level, any depth — straight in both, and in a flat picture too. Says nothing.

Above or below eye level, constant depth — straight in a scroll, symmetrically curved in a panorama. This is the test.

Above or below eye level, receding — curved in both, and the shapes differ: the panorama’s is symmetric about its extremum, the scroll’s is an asymmetric hyperbola running to an asymptote. A second test, harder to read, and the one that also rules out a flat perspective picture.

The distinction is therefore visible in the pictures rather than only in the models — which is where this field wanted to end — but it is visible on the line a reader is least likely to pick first, and the line they would pick is the one line that agrees.

The eye-level case is also not the only member of its family, which is worth knowing before trusting any single edge. A scroll draws straight every line lying in a plane that contains its own track: a line whose height above the eye grows in proportion to its depth is seen at one elevation from every position along the track, so it images along a single row. Checked on the model at a quarter of a radian of rise, seven samples return 42.500000 px apiece. Eye level is the horizontal member of that family and a rising road cut into a hillside beside the track is a tilted one, so a reader who finds one suspiciously straight receding edge should look for a second at a different rise before drawing any conclusion from it.

The conflation has a practical cost #

It would be easy to leave this as a taxonomic point, and it is not one. Treating a translating camera as a rotating one is a mistake with a price, and the price is measurable.

Stitching software builds a panorama by assuming the frames share a centre and finding the rotations that align them. If the camera actually translated — because the tripod head’s rotation point was not at the entrance pupil, or because the photographer turned on the spot rather than about the lens — the assumption fails, and it fails in a depth-dependent way. Near objects need one alignment and far ones need another, and no rotation satisfies both.

That is exactly the pushbroom’s failure at a smaller scale. A panoramic head offset by four centimetres is a camera whose eye travels four centimetres along an arc, and the resulting picture has no single centre by a miss proportional to that offset — which is the linear law this field’s third essay measured, with the track four centimetres long instead of twenty-six metres.

So the distinction between the two objects is the distinction between a panorama that stitches and one that does not, and photographers find the entrance pupil by trial precisely because the two look alike until a near object crosses a seam.