Visual Plane
every picture states the point it is correct from
The real instrument

The eye is a place, not a point

Rotate a camera about the wrong point and the sky still stitches perfectly while the foreground slides. The misregistration falls as one over the distance, exactly — which is what says the fault is the pivot and not the lens.
17 min read 7 figures Fitted, not assumedThe viewer is in the geometryOne machine used twice

Every essay on this site has treated the centre of projection as a point, which it is, and as an abstraction, which it is not. It is a place in the room, a few centimetres inside a lens, and knowing where it is decides whether two photographs can be joined.

Two frames stitched on the sky, with the pivot 50 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 3.2 px from where the other frame put it and the furthest 0.31 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 — 3.2 px out24 m — 0.3 px outthe sky registers to 1e-13 pxthe foreground does not — up to 3.2 px
Fig. 1 Two frames of a panorama, registered on the far field the way a stitcher registers them, with the near field left where it landed. The sky matches to 2e-13 px. The nearest post lands 3.2 px away from where the other frame put it, and the furthest 0.31 px.

The claim, and how it is tested

Two photographs taken from the same point, in different directions, are related by a homography — one that depends on the rotation between them and on nothing else. That is why a panorama can be stitched at all: the overlap can be brought into exact registration by a projective map, with no knowledge of the scene.

The word doing the work is same point. Move the camera between the two shots and the relationship is no longer a homography, because different parts of the scene move by different amounts — near things more than far things — and no map of the picture plane can encode that.

The figure runs the real procedure rather than staging a failure. The registration is computed the way a stitcher computes it: a homography fitted to four points at infinity, which is exactly the rotation homography. Then the near field is asked where it landed.

With the pivot at the entrance pupil, the answer is: exactly where it was, for every point at every distance, to 10⁻⁶ px. With the pivot 50 mm behind it, the nearest point is 6.3 px out.

The law the error obeys

The size of the error is not the interesting part. Its shape is.

The misregistration falls as 1/distance, exactlyAcross two decades of distance the stitch error times the distance is constant to 0.09%, 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.-0.50000.50010.50011.50log₁₀ distance to the point (m)log₁₀ misregistration after stitching (px)1.2 m → 7.85 px3.4 m → 2.77 px9.7 m → 0.97 px27.7 m → 0.34 pxpivot 50 mm behind the pupil, yaw 12°slope -0.9998
Fig. 2 Two decades of distance, on log axes. The slope is −0.9998 and the product of error and distance is constant to 0.09%.

Error times distance is constant. That is the signature of a parallax and it is what distinguishes this failure from every other one a stitch can have.

A misfitted focal length, a wrong distortion coefficient, a rotation estimated badly, a principal point in the wrong place: every one of those produces a misregistration that is a function of position in the image and does not care how far away the subject is. Only a translation of the centre of projection produces an error that falls as one over the distance, because only a translation makes the parallax angle depend on the depth.

So the 1/D law is a diagnostic. Photograph a scene with objects at several distances, stitch, and plot the residual against distance. Flat means the camera parameters are wrong. Sloping at −1 means the tripod head is.

Why the sky always looks fine

The most useful consequence of this arithmetic is a warning about how the failure presents.

The registration is fitted on whatever the stitcher can match, and in most panoramas that is the distant material: the horizon, the far buildings, the sky’s structure. Those are at effectively infinite distance, so their parallax is zero whatever the pivot offset, and they will register perfectly no matter how badly the head is set up.

Then the foreground — a railing, a table edge, the person standing nearby — is doubled or torn, and it looks like a stitching failure of some entirely different kind. It is not: it is the direct consequence of the pivot being wrong, showing up in the one place the fit did not look.

This is the same structure as several other findings on this site, and it is worth naming as a pattern: a fit that is exact on the data it was given, and wrong on the data it was not. A homography fitted to a façade’s four corners maps those four exactly and everything else wrongly when the façade is not planar. Four points fixing a rephotograph’s homography predict every other point exactly on a flat print and by 30.5 px on a rolled one. The residual on the points the fit was given is zero in all three cases and says nothing at all.

Why a homography and not a translation

A small point of mechanism that is easy to get backwards.

Two frames from one point differ by a rotation of the camera. It is tempting to think the overlap should therefore be related by a rotation of the image — shift one frame sideways by the amount the camera turned and they will line up.

They will not, except at the very centre. A rotation of the camera about its own centre changes the picture by the homography KRK−1K R K^{-1}, where KK carries the focal length and principal point and RR is the rotation. That map is a rotation of the image only in the limit of a very long lens and a very small angle. For anything wider it stretches the overlap, because the two picture planes are differently oriented and the projection from one to the other through the common centre is projective rather than rigid.

This is why simply sliding two frames over each other never quite works, and why every stitcher fits a projective map. It is also the same fact the rectification of a tilted photograph uses, and the same fact behind a photograph of a photograph being a homography — three quite different-looking operations that are one composition of projections through a common centre.

The figure builds the map from four world directions rather than from a general fit, precisely so that the map is constructed from the geometry rather than fitted to the data. Four correspondences determine a homography exactly; a fifth would make it a least-squares problem, and then the residual on the near field would be partly the fit’s own slack rather than the parallax.

Two frames stitched on the sky, with the pivot 100 mm behind the pupilThe far field registers to 2e-13 px, which is what makes the stitch look correct. The near field does not: the nearest post lands 6.4 px from where the other frame put it and the furthest 0.62 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 — 6.4 px out24 m — 0.6 px outthe sky registers to 2e-13 pxthe foreground does not — up to 6.4 px
Fig. 3 Twice the offset, and twice the misregistration at every distance. The sky is still exact, which is the half of the picture a stitcher is looking at.

Where the pupil is

The entrance pupil is the image of the aperture stop, formed by the lens elements in front of it, seen from the object side. It is where every ray of the picture appears to pass through.

Three things about it are worth knowing and none of them is obvious from the outside of a lens.

It is not the front element, the aperture ring, the sensor, or the tripod socket. On a retrofocus wide-angle it can sit in front of the front element, in the air. On a telephoto it can sit well behind the barrel.

It moves with focal length on a zoom, sometimes by several centimetres across the range, which is why a zoom used for panoramas has to be set to a marked focal length and left there.

And it moves slightly with focus, which is a second-order effect and is why panorama heads are set up at the focus distance the panorama will be shot at.

Nothing on the lens says where it is. It is a property of the optical design, it is occasionally in a manufacturer’s datasheet, and it is more often found by the test in the figure than looked up.

Finding it with the figure’s own instrument

The measurement in the figure is exactly the practical procedure, and the procedure is older than the arithmetic.

Set the camera on a rail so it can slide along its own axis. Line up two objects at very different distances — a near post against a distant chimney. Rotate the camera about the rail’s pivot, and watch whether the alignment holds.

At the pupil, it holds at every rotation. Anywhere else, the near object slides against the far one, and the direction of the slide says which way to move: slide the camera along the rail until the sliding stops.

The reason this works is one sentence. A ray of the picture passes through the pupil, and rotating a line about a point on it does not move the line. Rotate about any other point and the pupil translates, and a translated centre of projection is a genuinely different view of a three-dimensional scene.

That single sentence also explains the dome port, which is the same fact used constructively rather than diagnostically: a sphere centred on the pupil meets every ray of the picture along its own normal, because every ray passes through the centre.

What the offset costs, in practice

Some arithmetic to size the problem.

The eye translates by 2esin⁡(ψ/2)2e\sin(\psi/2) when the camera is rotated by ψ\psi about a pivot ee behind the pupil. For a 50 mm offset and a 12° step that is about 10 mm, and the parallax angle at a subject DD away is roughly 10 mm/D10\,\text{mm}/D. On a 690 px frame with a 44° lens the focal length is about 854 px, so the misregistration in pixels is about 8.5/D8.5/D with DD in metres.

Which gives: 6 px at 1.5 m, 3 px at 3 m, 1.5 px at 6 m, and under a pixel past 10 m.

So the rule of thumb that falls out is that the pivot matters when there is anything within a few metres, and does not otherwise. A panorama of a landscape from a hilltop can be shot on any tripod at all. A panorama of a room cannot, and the difference between the two is entirely the presence of near material.

That is why interior panoramas — estate agents’ room tours, virtual tours of buildings — are the case where panorama heads are not optional. Everything in the frame is within a few metres, so everything in the frame carries parallax.

The nodal point, and why the name is wrong

Panorama heads are sold as “nodal heads” and the point being sought is usually called the nodal point. It is not.

A lens has two nodal points, and they are a real and different thing: the pair of points such that a ray aimed at the first emerges from the second travelling parallel to its original direction. They matter for computing where an image forms and how a lens behaves when it is tilted, and they are properties of the refracting power of the system.

The point that must be on the axis of rotation for a panorama is the entrance pupil — the point every ray of the picture appears to pass through, which is the image of the aperture stop formed by the elements in front of it. For a thin lens focused at infinity the two coincide, which is presumably where the confusion started, and for a real retrofocus wide-angle they can be centimetres apart.

The mistake matters because it sends people to the wrong number in a datasheet. A lens whose rear nodal point is published, set up on that number, will be off by however far the pupil sits from it — and the parallax test will show it immediately, which is the practical argument for measuring rather than reading.

There is a cleaner name in use in the panorama community, no-parallax point, and it has the advantage of being defined by the test rather than by the optics. This site prefers “entrance pupil” because that is what it is, and mentions the other two so that a reader meeting them elsewhere knows which is which.

The exact case, and why it is exact

The figure asserts two things about the centred case and the second is the one that makes it a result rather than a set-up.

The far field registers to 2e-13 px — that is the homography doing what a homography does, and it would be true at any pivot offset.

The near field registers to 10⁻⁶ px as well, at every distance from 1.5 m to 24 m. That is the claim: with the pivot at the pupil, there is no residual parallax at any depth, because the two frames really are two views from one point and the rotation homography really is the exact relationship between them.

A check that only tested the far field would pass at every offset and measure nothing, which is the trap this site keeps finding: a cross-ratio evaluated at four consecutive divisions, a conformality test taken along the surface’s own axes. Both were necessary conditions evaluated at the one input where they cannot fail. Testing the stitch on the sky alone would be the third.

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 panoramas are shot in pieces at all: the flat picture surface runs away as the field widens, so a wide enough view cannot be a single rectilinear frame. Every argument in this essay exists because the alternative is to take several pictures and join them.

The connection to the panorama field

What a 360 photograph is treats a panorama as a record of the directions from one eye at one instant — a stored light field of a single point, projected onto whatever surface is convenient afterwards.

This essay is the condition on that sentence. A panorama is a record of directions from one point only if the camera actually rotated about one point. If it did not, the object being stored is not a set of directions from a place; it is a set of directions from a small circle of places, and no picture surface will reconcile them.

The failure has a name in the panorama literature — parallax error — and it is usually described as a stitching artefact. It is better understood as a category error. The stitched result is not a bad panorama; it is not a panorama at all, in the sense that there is no viewpoint from which it is a correct projection, which puts it in the same class as a picture taken through water.

One room at 200° 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 cannot hold this field of view at all; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.no picture at 200°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%200° across in every panelsame scene, same angle, six surfaces
Fig. 5 What the registration is in service of. A panorama is a record of the directions from one eye at one instant, cast onto whatever surface is convenient afterwards — and this essay is the condition on the word one.

Deliberate parallax, which is the same instrument

Everything above treats the translation of the centre as a fault. Move the camera on purpose and the same quantity becomes the most useful measurement in photogrammetry.

Two views from two known positions give every scene point a disparity, and the disparity is fB/DfB/D for a baseline BB — which is the same 1/D1/D law, read as a signal rather than as an error. That is stereo, and it is how depth is recovered for every point of a scene rather than for the identified planes single-view metrology is restricted to.

What a second view does not buy is the one thing a single view cannot supply either. The baseline has to be known in real units, and if it is not, the recovered scene is determined up to a single overall scale — the same ambiguity, inherited rather than removed. Two views give shape and no size, exactly as one view gives ratios and no size.

So this essay sits at a boundary the site has not crossed. The pivot offset here is a nuisance to be eliminated; treated as a known baseline it becomes an instrument, and the geometry of two views is a subject in its own right. That subject is the natural next ground for this site and it belongs to a later phase — the argument for putting it off is that the multi-view case deserves a field rather than an essay, and that the field it would deepen is the one this essay has just finished borrowing from.

The cost of assuming the principal pointThe two-vanishing-point route to a focal length needs a principal point supplied, and every textbook supplies the middle of the frame. On a shifted or cropped picture that is wrong, and f² = −(v₁ − p)·(v₂ − p) turns the error into a product of two large numbers: 8.0% at 150 px of shift. The three-point recovery does not assume it and has no such error.02468050100150how far the principal point really is from the middle of the frame (px)error in the focal length, from assuming it is not (%)unshifted: the assumption is truea 84° lens7.98% at 150 px of shift
Fig. 6 Another intrinsic that has to be found rather than assumed. The pupil is where the projection’s centre physically is; the principal point is where its axis meets the sensor, and neither is written on the lens.
A case front 40 mm thick, with and without itEvery point has moved — by up to 7.3 px — and every vanishing point has not, to 7e-6 px. So the camera recovered from this picture's own vanishing points is the camera that took it, to 7e-10 relative, out of a picture in which nothing is where it was.no single viewpoint — the rays miss by 7.3 px, depth-dependentf recovered from it: 396.88 px
Fig. 7 A translation of the centre of projection is the one thing no map of the picture plane can encode. A slab, by contrast, moves points and leaves directions — which is why a picture through glass still stitches.

Closing the field

This field opened with a lens bending straight lines and closes with the centre of projection turning out to be a place with a position that has to be found.

The four measurements between them are a reasonable summary of what separates a real camera from the pinhole this site is built on. It bends straight lines, by 17.8 px on a wide frame at a coefficient a phone would be embarrassed by. It destroys the invariant, by 1.29% where the pinhole is exact to 2e-16. Its principal point is not the middle of its picture, and assuming it is costs 1.6% of the focal length. And its centre of projection is a place, 50 mm from which is enough to tear a foreground apart.

Every one of the four is recoverable from a picture: the distortion from straightness, the principal point from three vanishing points, the pupil from a parallax test. That is the thread this field shares with the site’s foundation — the camera recovered from the picture it drew — extended from the ideal camera to the real one, and it holds up: nothing about a real camera has to be looked up if the picture is available.

Where this goes next

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. 00.50011.50205101520the dome's centre, off the entrance pupil (mm)worst departure from the pinhole it would be in air (degrees)centred: exactly zero6 mm → 0.635°a 100 mm dome in acrylic, n = 1.4910.106° per mm of centring error The port that is not there A flat window into water costs a lens a third of its field. A sphere centred on the entrance pupil costs nothing at all — not nearly nothing, exactly nothing — and six millimetres off centre costs 0.635°. level — the top is cut off0.00° of spreadtilted 13°4.55° of spreadshifted 95 px0.00° of spreada shift moves every point by exactly the shift95.0 px, and no direction at all The principal point is not the centre Every textbook that computes a focal length from two vanishing points supplies the middle of the frame as the principal point. On a shifted or cropped picture that is wrong, and it costs one and a half per cent of the answer at a fifth of a frame's shift.