Visual Plane
every picture states the point it is correct from
The eye that moves

A scroll is not a panorama

Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.
17 min read 11 figures Not a projection at allThe surface is a choice

Worth reading first: What a 360-degree photograph actually is · A scroll is a camera that moves.

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.

Curving straight lines and having no centre are two different thingsThe rms miss of the best single centre, for two cameras that both draw straight world lines as curves. A rotating eye keeps its centre exactly — 2e-15 m, which is the solver's noise floor. A translating eye has none: 7.97 m over 27 m of track. A panorama is a projection and a scroll is not, and no amount of looking at the curves tells them apart.a rotating eye — the panorama2e-15 ma projectiona translating eye — the scroll7.97 mnot oneboth of these draw a straight world line as a curverms miss of the least-squares centreone of them is a projection
Fig. 1 The measurement that separates them: the rms miss of the best single centre, fitted from each system’s own rays. A rotating eye keeps its centre at 2e-15 m, which is the solver’s noise floor. A translating eye misses by 7.97 m over 26 m of track. Both curve their straight lines; only one of them is a projection of anything.

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.

One room at 120° across, cast onto six picture surfacesEach panel holds the same angular width of the same scene. The flat plane keeps every straight line straight and pays for it at the edges; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces
Fig. 2 The panorama’s kind of failure: a projection from one point onto a surface that is not flat, measured on the three things a picture surface can do to the world. Every one of these still has a station point. A scroll appears on none of these panels because a scroll is not a choice of surface — it is a choice about the eye.

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 1.9×10−151.9 \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.

Two frames stitched on the sky, with the pivot 40 mm behind the pupilThe far field registers to 1e-13 px, which is what makes the stitch look correct. The near field does not: the nearest post lands 2.6 px from where the other frame put it and the furthest 0.25 px, an error falling as 1/distance — the signature of a parallax rather than a calibration error.the far field — where the stitch was fitted2.2 m — 2.6 px out24 m — 0.2 px outthe sky registers to 1e-13 pxthe foreground does not — up to 2.6 px
Fig. 3 The same distinction as a practical failure. A panoramic head rotated about the wrong point is a translating camera by a few centimetres, and the stitch error that follows is depth-dependent and cannot be removed by any alignment. A handscroll is that failure taken to metres and made the point of the object.

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.

How wide the picture gets as the field of view opensOn a flat plane the picture's half-width is tan(θ/2): it multiplies by 6.6 between 120° and 170° and is unbounded at 180°. On a cylinder it multiplies by 1.42 over the same range and keeps going past 180° without incident.0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°
Fig. 4 Why the flat plane is not an option for either object: a flat picture’s half-width is the tangent of the half-angle and is unbounded at 180°, while a cylinder’s keeps going past it. A panorama solves this by curving the surface and a scroll by moving the eye, and the two solutions have almost nothing else in common.

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.

6 pictures, one track and one scene, recovered togetherEvery point read to 1 px. Bundle adjustment moves all 168 parameters at once and lands at 0.332 px of reprojection error in 5 iterations. The recovered camera centres, drawn over the true ones after the similarity the pictures cannot fix is removed, agree to 2.0e-3 of the track's own mean radius. Nothing in the solve was ever told where a camera or a point was.view 1view 6the scene, in plan — recovered points and cameras over the true onesreprojection 0.332 px · track 2.0e-3264 observations, 168 parameters
Fig. 5 What happens when both the position and the direction are free: the fourth cell of the table, which is not a drawing system but a reconstruction problem. The camera track and the scene are recovered together, from many pictures, and the result is a model rather than a picture. A scroll is what the same freedom looks like when it is constrained to a line and drawn on silk.

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.

Area scale across the picture, for six surfacesThe flat plane's area scale is sec³θ and leaves the frame before 70°. The equal-area fisheye is the flat line at 1, which is what its name asserts and what this measures.024680204060angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame
Fig. 6 Area scale across the picture for six surfaces, in the field that measures it. The flat plane’s runs away and leaves the frame; the equal-area fisheye is the flat line its name asserts. A scroll appears on none of these, because its coordinate is a distance rather than an angle — a third answer this figure has no row for.

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, it has a horizontal asymptote and a vertical one, and the lines it draws straight are the ones at constant depth rather than the ones at eye level.

So the two families are separable from a single well-chosen line, and the choice is what makes it work: find a long straight edge running at eye level and away from the viewer. A panorama draws it straight; a scroll bends it. Find one running at eye level and across; both draw it straight. Find one below eye level and receding; both bend it, in different shapes.

That is a satisfying place for this field to end, because it means the distinction is not merely a fact about the models. It is visible in the pictures, to anybody who knows which line to look at.

Five great circles, imaged on the stereographicEach is fitted as a general conic and comes back a circle: |A−C|+|B| is 1e-9 of the fit's own scale. Stereographic is the only surface here that does this.fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples
Fig. 7 The panorama’s signature, drawn: great circles imaged on a curved surface, each one a conic that the fit recovers exactly. A scroll’s hyperbola is asymmetric and is straight when the line holds its depth. Two curved images of a straight line, distinguishable from the curve alone.

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.

The misregistration falls as 1/distance, exactlyAcross two decades of distance the stitch error times the distance is constant to 0.07%, and the slope on log axes is -0.9998. That is what says the fault is the pivot and not the lens: a calibration error would not care how far away the subject is.-1-0.50000.5000.50011.50log₁₀ distance to the point (m)log₁₀ misregistration after stitching (px)1.2 m → 6.28 px3.4 m → 2.21 px9.7 m → 0.78 px27.7 m → 0.27 pxpivot 40 mm behind the pupil, yaw 12°slope -0.9998
Fig. 8 The cost, as a curve. Stitch error against the distance of the object being aligned, for a camera rotated about the wrong point: it falls as 1/D and never reaches zero, so there is no alignment that serves near and far together. A handscroll is this failure with the offset made large and made the point.
What each drawing system preservesEvery cell is read out of the system's own map rather than typed: the centre is fitted from its rays, measure is the site's own midpoint test, diminution is the drawn size of a fixed object at two depths, bounded depth asks whether infinite depth lands in finite page, and straightness is the sag of a receding line. Only perspective has a centre, and it is the one system with no true measure. No row has both.a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationfilled means the system keeps itmeasured from each system's own projectionno row is filled in both of the first two
Fig. 9 And the summary the next field builds on. The first column is a centre, fitted from each system’s own rays, and it is the column this essay is about: a rotating eye keeps one and a translating eye does not, whatever their pictures look like.

The pair, once more

The rays of 16 m of scroll, and the point they miss by 5.21 mPlan of one section. Each ray leaves the eye at its own column, so the eyes lie along a track rather than at a point. The circle is the least-squares centre drawn at the radius of its own miss — 5.21 m, which the closed form puts at 5.21 m, the standard deviation of a track that long. A single column of the same scroll fits exactly.the best point, missed by 5.21 m16 m of the eye's trackno single viewpoint — the rays miss by 5.21 mthe eyes are a track, not a point
Fig. 10 A translating eye’s rays, and the point they miss. The circle is drawn at the radius of its own miss, which is the honest way to draw a point that is not there.
Five great circles, imaged on the stereographicEach is fitted as a general conic and comes back a circle: |A−C|+|B| is 1e-9 of the fit's own scale. Stereographic is the only surface here that does this.fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples
Fig. 11 A rotating eye’s, on the surface that curves them: five great circles, each fitted as a conic and each coming back a circle. Every ray in this picture passes through one point and the curve is the surface’s doing. Two pictures, both curved, and only one has somewhere to stand.

Where this goes next

fitted conic: 1.467 from being a circlecylinderthe samples, fitted The cylinder, and the price of going all the way round A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices. the eye's trackthe eye at x = -10.5 mevery point is drawn by the one position of the eye that is level with itplan — the eye's track and the scans it makes28 m of travel A scroll is a camera that moves A Chinese handscroll is not a picture with a wandering viewpoint or a picture with no viewpoint. It is the image of an eye that travels along a track and records one vertical line at a time, and that object has an exact geometry — orthographic along the roll, perspective across it. no picture at 300°the plane is unbounded at 180°plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%300° across in every panelsame scene, same angle, six surfaces What a 360-degree photograph actually is The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

centre of projectionCylindrical projectionEquirectangularHandscrollMoving viewpointPanoramaPicture surfacePushbroomSpherical panoramaStation point