The pivot that is not the eye

Every panorama in this field so far has been a surface. The cylinder is a map from a direction to a mark; so is the equirectangular rectangle, and so are the four others this collection carries. Each of them is a perfectly good picture surface and each of them quietly assumes the thing this essay is about, which is that all those directions were taken from one place.

An instrument does not oblige. A camera is bolted to a tripod by the screw under its baseplate, and the light crosses the lens somewhere in front of that. Turn the camera and the pupil goes round a small circle.

One line of geometry, and it is exact #

Put the pivot at the origin and the entrance pupil a distance $e$ in front of it, along the optical axis. A ray leaving the pupil at an angle $\gamma$ from that axis is the line through $e\hat{f}$ in the direction $\hat{d}$, and its distance from the pivot is a cross product:

$$\text{miss}(\gamma) \;=\; \left| e\,\hat{f} \times \hat{d} \right| \;=\; e \sin\gamma$$

That is the whole of it. Nothing else is in it — not the scene, not the distance to anything, not the focal length, not which way the camera is pointing.

The two routes matter because the site’s habit is that a figure quoting only algebra is quoting algebra. missAt writes $e\sin\gamma$ down; measuredMiss builds the ray and drops a perpendicular onto it from the pivot. They are not the same computation and they have to give the same number.

Three consequences fall straight out and each of them is a surprise the first time.

The centre column of every frame is exact #

At $\gamma = 0$ the miss is zero. A ray along a frame’s own optical axis leaves the pupil at $e\hat{f}$ and travels along $\hat{f}$, so it is the radial line through the pivot. It passes through the pivot exactly, and it does so whatever $e$ is.

So a panorama assembled from infinitely narrow strips — one column of pixels per frame, taken at the centre of each — is a genuine central projection. Not approximately: exactly. And its centre is the pivot, which is a point where no lens has ever been.

That is a strange thing to be true. The photographer’s error is measured in millimetres of hardware, and the picture it produces — if the strips are narrow enough — is a correct picture from a place six centimetres behind the glass.

It is also the reason the offset is not fatal by nature. A defect that vanishes in a limit is a defect with a price on it, and the rest of the field is about what that price is.

The rays are tangent to a circle, so the picture has a radius #

For a strip of finite width the rays do not meet. Every one of them passes the pivot at $e\sin\gamma$, and the family of them is tangent to a sphere of that radius. Draw the horizon row alone and it is a circle; take the whole strip and it is a sphere.

That is the honest replacement for a centre of projection. A perspective picture is a projection from a point. A panorama stitched from a camera on the wrong pivot is a projection from a ball, and its radius is a number the arrangement fixes.

The rest of this site has met the same shape of failure four times, and it is worth putting the five side by side because they are not variations of one thing.

A picture through water has no centre because the medium bends the light. A scroll has none because the eye travels. A curved mirror has none because the reflector turns with position. A picture with two eyes in it has none because it was drawn from two on purpose. This one has none because the instrument was screwed to its mount in the ordinary way.

The ghost a reader can actually see #

None of the above is visible in a photograph. What is visible is the seam: the place where two frames were joined, where a railing appears twice and a lamp-post has a kink in it.

A stitcher registers frames on their far field, which is what a homography through four points at infinity does. That registration is exact for anything at infinity and wrong for everything else, because the two frames were taken from two places. The residual is a parallax.

The flatness of that second curve is the whole argument. A misregistration that did not fall off with distance would be a calibration error — a lens whose focal length was wrong, a rotation that was mismeasured. One that falls as $1/D$ is two eyes in different places, and there is nothing else it can be.

This is the same instrument the lens field uses on a plumb line: a residual is not a number, it is a shape, and the shape says which model is missing.

What the strip width buys #

The seam sits half a frame-spacing from each frame’s axis. Stitch $n$ frames evenly round a turn and each contributes the strip $|\alpha| \le \pi/n$, so the worst horizontal miss is

$$e \sin(\pi/n)$$

and it halves when the frame count doubles. The photographer who shoots more frames is buying exactly this.

The two halves are one number #

The tangent sphere and the seam ghost are described above as two consequences. They are one, and joining them turns the essay’s geometry into a number a photographer can act on.

The two frames meeting at a seam have their pupils at $\pm\pi/n$ around the pivot circle, so those two pupils are separated by a chord of $2e\sin(\pi/n)$ — twice the worst miss, and lying perpendicular to the seam’s own bearing. Two eyes that far apart looking at something $D$ away disagree by $2e\sin(\pi/n)/D$ radians, which on a camera of focal length $f$ pixels is a doubling of

$$\text{ghost} \;=\; \frac{2\,e\,f\,\sin(\pi/n)}{D} \ \text{pixels}.$$

So the ball the picture is projected from and the railing drawn twice are the same fact measured in different units: the ball’s radius is the parallax baseline, and the ghost is what a baseline of that size does at that distance. The $1/D$ the seam figure reports is in the expression, and so is the halving with frame count, and so is the exactness of the centre column — at $n \to \infty$ the sine goes to zero and both vanish together.

Put numbers in it and the practical picture changes shape. A 60 mm offset, six frames round the turn, a foreground at two metres, and a lens of about 950 pixels of focal length gives 28 pixels of doubling, which is a railing visibly drawn twice. Requiring under one pixel needs

$$e\sin(\pi/n) \;<\; \frac{D}{2f},$$

which for those numbers is 1.06 mm — so at six frames the pivot has to be right to two millimetres, which is the tolerance the cylindrical panorama arrives at from the single-offset case and is the reason the no-parallax point is worth finding rather than estimating.

The same inequality settles what more frames can and cannot rescue, and the answer is bracing. Holding the 60 mm error and asking how many frames would bring the ghost under a pixel gives $\sin(\pi/n) < 1.06/60$, that is $n > 178$. A hundred and seventy-nine frames, to make up for six centimetres of hardware, on a turn that six frames would otherwise cover.

Two readings, and they pull in opposite directions.

Frame count is a weak remedy and a real one. It buys a factor of $n$, linearly, and linear is not much against an error of the wrong order — but it is the only remedy available after the photographs have been taken, since nothing in software can move a pupil that has already been somewhere. A stitch of many narrow strips from a badly mounted camera is genuinely better than a stitch of few wide ones, and the improvement is exactly proportional.

And the tolerance is a fact about the scene, not about the rig. $D$ is in the numerator of the requirement, so the same camera on the same misaligned mount is faultless for a landscape and hopeless for a room. The photographer who has never seen a ghost has not calibrated anything; they have photographed distant things.

The expression also says which of the two remedies is worth the effort in a given case, which is a division. Doubling the frame count and halving the offset buy the same factor, so the question is only which is cheaper — and moving a camera back on a rail is a one-time adjustment measured in millimetres while doubling the frames is paid on every panorama for ever. The pivot is the thing to fix, and the strip width is what to do when it is too late.

One last reading, because it explains a practice that looks like superstition. A slit-scan panoramic camera exposes one narrow column at a time while the body turns, which is the $n \to \infty$ limit made mechanical: the sine goes to zero, the tangent sphere collapses to the pivot, and the picture is a true central projection from a point inside the camera whatever the lens’s pupil is doing. Such a camera therefore needs no no-parallax adjustment at all, and photographers who have used one report exactly that and are usually not believed. The expression above says they are right, and says why the same freedom is unavailable to anybody assembling a panorama out of whole frames.

Which raises the obvious question and it has a non-obvious answer. Shoot enough frames and the ghost goes to nothing; so is the whole problem a matter of shooting more? No, and the reason is the other angle in $\gamma$. That is the next rung and it is the part of this that nobody says out loud.

Why the miss is a sine and not a distance #

It is worth pausing on the form of the law, because it is the reason the whole thing is tractable.

The miss does not depend on how far away anything is. It is a property of the ray, not of what the ray hits, so it can be computed before a scene exists — and it can be quoted as a length in the room rather than as a disagreement about a photograph. That is the same move the refraction field makes when it reports millimetres of world rather than pixels of picture: a picture-space residual mixes the defect with the focal length, and a world-space one does not.

It also means the offset and the angle are separable. Doubling $e$ doubles every miss; changing the strip width changes the sine and nothing else. Two knobs, one product, and neither of them is the scene.

The two circles are not the same circle #

The plan view has two circles in it and they are worth separating, because conflating them is the natural mistake.

The pupil circle has radius $e$. It is where the lens actually went, and it is the hardware error: sixty millimetres of rail, measured with a ruler on a workbench.

The tangent circle has radius $e\sin(\pi/n)$. It is what the picture has instead of a centre, and it is smaller — by the sine of half a frame spacing, which at six frames is a half. So the defect in the picture is not the defect in the mounting; it is the mounting’s error multiplied by a number the photographer chose when deciding how many frames to shoot.

That separation is the useful one. A reader with a photograph and a complaint wants to know which of the two to change, and the answer is that they multiply — so halving either halves the result, and the cheaper one to halve is usually the count.

What a stitcher could do instead, and what it would need #

The obvious repair is to stop registering on the far field and register on the scene: warp each frame by a map that depends on how far away each piece of it is, so that near things line up too.

That works, and the reason it is not what stitching software does is instructive. The map is a function of depth, and a pair of photographs from two places does not carry depth for nothing — it has to be recovered, from correspondences, with all the machinery the two-view field sets out and all the failures it prices. A stitcher that did this would be a stereo reconstruction with a panorama on the end of it.

So the far-field registration is not laziness. It is the choice to make a picture out of an assumption that is exactly true for everything at infinity and false by a computable amount for everything else — and the computable amount is what this essay is.

The refusal, and what it protects #

A panorama needs at least two frames, and stitchMiss refuses one. That looks like housekeeping and is not quite: the single-frame case is a perfectly good photograph with a perfectly good centre — its own pupil — and calling it a panorama with a pivot error would be attaching a defect to a picture that does not have one.

The distinction the refusal enforces is that the offset only becomes a defect when frames are combined. One frame from a badly-mounted camera is exact. Two are not, and the amount by which they are not is a property of the join rather than of either.

The control, which is the whole reason the numbers mean anything #

Set the offset to zero and every ray of every frame passes through the pivot exactly. The least-squares fit lands on it at the arithmetic floor — not small, zero — and the tangent circle has no radius because there is nothing for it to be tangent to.

That figure is worth looking at twice, because it is the one place on this site where the no-viewpoint strip and the viewing-distance strip are drawn by the same generator at two ends of one slider. The exemption belongs to the configuration rather than to the name — which is how the rolling shutter and the curved mirror are treated, and for the same reason.

The tripod rail, priced #

The practical form of all this is a rail: a bracket that slides the camera back over its mount until the pupil sits on the axis. Panoramic heads have had one for a century and the instructions never say what it is worth.

It is worth $e\sin\gamma$, and that is a length in the room. At six centimetres of error and six frames the strip’s edges miss the pivot by three centimetres; the whole bundle, top of frame to bottom, misses its own best point by two and a half. Whether that matters is a question about the scene — a landscape at a hundred metres does not care, a table two metres away does — and the useful thing is that the scene enters only at the last step, through a division.

What this does not say #

It does not say a stitched panorama is not a picture. It is a picture; it is not a projection, which is a narrower claim and is the one this site makes about everything.

It does not say the offset is the largest error in a panorama. The lens is usually larger, and the lens field measures it; so is the assumption that the rotation was about a fixed axis at all.

And it does not say the pivot is a good centre of projection to use. It is the centre, in the narrow-strip limit, and a reader wanting the picture’s viewing distance should take it from there. But the strips are not narrow, and how far from narrow they are is the subject of the next rung.

A defect that vanishes in a limit has a price rather than a verdict. Find the limit first, then measure how far the instrument is from it — the two numbers together are a design, and either one alone is a complaint.

The limit here is a slit, and the instrument that actually takes it is a real camera with a real name. That one has no parallax at all and pays for it somewhere else entirely, which is two rungs along.