Focusing moves the pivot past its best place

The entrance pupil walks with the angle finds that the place a picture is a projection from is not a point in a wide-angle design: chief rays traced through a strongly curved front element cross the axis 15.07 millimetres behind its front vertex near the axis and 4.23 millimetres nearer the front at eighty degrees of field. So no pivot makes a wide panorama seam clean, and the best one still leaves 1.41 arcminutes at one metre.

Every one of those numbers is measured with the lens at one focus. Moving the focus moves the pupil too, and the question is whether that movement is a correction on top of the walk or the larger of the two.

The travel is a closed form #

For the simplest kind of focusing — the whole lens moving away from the sensor as one piece — the entrance pupil travels with the glass. Its displacement relative to anything bolted to the camera body, which is what a tripod mount and a panorama head are, is therefore exactly the extension: how much further the lens stands from the sensor than its own focal length.

That is the focal length squared, divided by the subject distance less the focal length. For a fifty-millimetre lens it is a quarter of a millimetre at ten metres, 1.28 at two, 2.63 at one, and 5.56 at half. Focusing is a zoom measures the same extension from the other side — it is what makes a lens focused close cover a narrower angle than the same lens at infinity, and its 5.56 millimetres at half a metre is this 5.56.

It overtakes the walk at half a metre #

The comparison the previous measurement leaves open is with the walk, and it needs both quantities measured over the same field or it is not a comparison.

Over the sixty degrees of image angle the seam measurement traces, the pupil of that front group walks 5.53 millimetres. The extension reaches that at a subject 502 millimetres away.

So the two quantities cross at about half a metre, and the crossing is what decides which one a rig has to worry about. For anything further than half a metre the pupil’s position is decided mostly by which part of the field a ray comes from; for anything nearer it is decided mostly by where the lens is focused. Neither is negligible in the region around it, which is most of the distances a panorama of a room is shot at.

The crossing moves with the focal length, and quickly: it is 170 millimetres for a twenty-eight millimetre lens and 3.4 metres for a hundred and thirty-five. A long lens focused at a few metres has moved its pupil further than any field-angle effect could, and a wide one has not.

What a head aligned at the wrong focus costs #

A panorama head is aligned once, on a bench, with two marks at different distances lined up as the camera swings. The alignment is done at whatever focus the lens happened to be at, and then the panorama is shot at whatever distance the room is.

Take the alignment done at infinity and the panorama shot at a metre. The pupil has moved 2.63 millimetres forward and the pivot has not, so the rig is pivoting 2.63 millimetres behind the entrance pupil. Along a seam between two frames sixty degrees apart, with the nearest subject at a metre, the worst disagreement is 7.46 arcminutes — against 4.39 for a rig pivoting at the paraxial pupil, and 1.41 at the best fixed pivot.

At a subject half a metre away the same alignment leaves 17.45 arcminutes, which at sixty pixels to the degree is seventeen pixels of misregistration along the seam.

The error does not grow from zero #

The obvious model of this is that aligning at the wrong focus introduces an error which grows with how wrong the focus is, and that model is wrong in an instructive way.

Aligned at eight metres and shot at one, the rig leaves 3.31 arcminutes. At four metres, 2.21. At two, 2.84. At one, 7.46. So the error falls and then rises, and its minimum is at an alignment distance of about three metres — where it reaches 1.41 arcminutes, which is the best a fixed pivot can do at all.

The reason is that the best pivot is not the entrance pupil. It is 0.867 millimetres behind the paraxial pupil, at the midpoint of the pupil’s own walk across the field, and an alignment at the wrong focus moves the pivot backwards by exactly the extension. A little of that motion carries the pivot toward the best place; too much carries it past.

So a head aligned at infinity and used at a metre is not merely misaligned — it has overshot the optimum by 1.77 millimetres, having passed through it on the way.

The arithmetic that turns millimetres into arcminutes #

The conversion between the two currencies is one line, and having it makes every number above predictable rather than reported.

A pivot a distance $\delta$ from the pupil, swinging between two frames whose axes are $2h$ apart, photographs a subject at distance $D$ from two centres separated by about $2\delta\sin h$. The angle that separation subtends at the subject is the misregistration, so

$$\text{seam} \;\approx\; \frac{2\delta\sin h}{D}.$$

At $\delta = 2.63$ mm, $h = 30°$ and $D = 1$ m that is 2.63 milliradians, which is 9.0 arcminutes — against the 7.46 the trace gives, the difference being that the walk moves the effective $\delta$ across the field rather than holding it.

Three things follow from that one expression and they are what a photographer would want.

The error is linear in the offset, so halving the misalignment halves the seam. It is inversely proportional to the subject distance, so the same rig that is unusable on a table top is fine on a landscape — which is why panorama heads matter indoors and not outdoors. And it grows with the frame spacing, so a panorama shot with more overlap is more forgiving of a wrong pivot as well as easier to stitch.

What the pupil is, before it moves #

All of this treats the entrance pupil as the thing to pivot about, which the hole a scene actually sees establishes rather than assumes: the aperture stop is not the centre of projection, and modelling a fifty-millimetre lens with its stop eighteen millimetres behind the glass puts the chief rays from every object distance through one point 28.1 millimetres on the other side of the lens, to 3.6 × 10⁻¹⁵ millimetres.

That point is what focusing carries, and it is worth noticing that the two measurements are about different failures of the same idealisation. The pupil exists as a point at one focus and near the axis, and the two ways it stops being one are independent: a field angle spreads it across the aperture, and a focus change translates it bodily.

The rest of the panorama subject inherits both. The camera that is a cylinder treats a rotating camera as one instrument rather than as a sequence, which is exact only if the pivot is on the pupil; a rig is right on one surface measures the depth at which a stitched rig is correct, which is where the parallax a wrong pivot leaves happens to vanish. Both take the pivot as a fixed property of the rig, and the measurement here says it is a property of the rig and the focus ring together. The pivot that is not the eye is the same distinction drawn for a rotating camera without a lens model in it, and the parallax a rig cannot shoot away is the part of the error no pivot removes at all.

Which makes the folklore not quite right #

The standing advice, which the eye is a place, not a point records, is to set a panorama head up at the focus distance the panorama will be shot at. That advice is better than nothing and it is not the best available.

Aligning at one metre and shooting at one metre puts the pivot at the entrance pupil for that focus, which leaves 4.39 arcminutes. Aligning at about three metres and shooting at one leaves 1.41 — three times better, and it is the best a single pivot achieves.

The reason the folklore is stated the way it is, is that it is aiming at the pupil, and the pupil is the right target only if the pupil is a point. It is not: it walks across the field, and the pivot that minimises the worst seam sits at the middle of that walk rather than at either end of it. An alignment at a longer focus than the shooting distance is a way of reaching the middle with a procedure that only knows how to find the end.

That is a thoroughly lens-specific number and it should be said so. Three metres is the answer for this front group and this frame spacing; another design walks a different distance and wants a different alignment focus. What transfers is the shape: align at a longer focus than the panorama is shot at, by an amount that puts the pivot at the middle of the walk rather than at the near-axis pupil.

How to align a head, given this #

The procedure a photographer actually follows can be stated in a way that uses the result rather than working around it, and it costs no extra equipment.

The standard alignment lines up a near mark and a far mark and turns the camera until they stay lined up. That finds the pupil at whatever focus the lens is set to, because it is the point about which a rotation produces no parallax at all — and it finds it at whatever field angle the two marks sit at, which is usually near the middle of the frame where the near-axis pupil is.

Two changes follow from the measurements above. Put the marks near the edge of the frame rather than the middle, so the alignment finds the pupil at a field angle nearer the middle of its walk rather than at the near-axis end. And set the focus further than the panorama will be shot at, by enough to carry the pivot the rest of the way. For this front group at a subject of a metre the second change alone is worth a factor of three, from 4.39 arcminutes to 1.41.

Both are corrections toward the same place, which is the pivot at the middle of the pupil’s own walk, and neither needs the walk to be known: the first reaches it optically and the second reaches it mechanically. A photographer who does both has over-corrected, which is a reason to do one of them and check.

The check is the same one the alignment uses. Swing the camera through the full spacing rather than a few degrees, with the near mark at the edge of the frame rather than in the middle, and look at the near mark’s drift. That is the quantity the seam will show, and it is the only one worth minimising.

Two motions, one pivot #

It is worth setting the two displacements beside each other, because they are different kinds of thing and a rig has to deal with both.

The walk with field angle is a property of the optical design at a fixed focus, and it happens within one exposure: different parts of one frame are projected from different points, so no pivot can be right for all of them at once. That is why the best pivot leaves 1.41 arcminutes rather than zero.

The travel with focus is a rigid displacement of the whole pupil, the same for every part of the frame. It can be corrected exactly, by moving the pivot — and it is the only one of the two that can.

So the practical division is clean. The walk sets a floor no rig gets below; the focus travel is an error a rig can eliminate, and a rig that does not eliminate it is usually carrying an error several times the floor. At a subject one metre away the floor is 1.41 arcminutes and an infinity alignment contributes 7.46.

What this does not settle #

Only unit focusing is measured. A lens focused by moving one internal group — which most modern lenses are — does not carry its pupil with the glass in any simple way, and the displacement can be smaller, larger, or of the other sign. The closed form here is exactly right for the simplest mechanism and is a rough guide for nothing else.

One front group, one frame spacing, one subject distance. The 1.41 arcminutes and the three-metre alignment distance belong to the meniscus group traced here with frames sixty degrees apart and the subject at a metre. The entrance pupil walks with the angle already shows the walk differs between designs, and the alignment advice follows the walk.

The frames are taken as identical apart from the rotation. A lens refocused between frames also changes its angle of view, which focusing is a zoom measures at 5.56 millimetres of extension for a fifty-millimetre lens at half a metre, and a stitcher asked to join frames of slightly different coverage has a second problem on top of the parallax.

And the seam is measured as an angle rather than as a stitch. What a stitcher actually does with a misregistered seam — blend it, warp it, choose a cut line through it — decides how visible a given number of arcminutes is, and that is not a geometry question.

Still open: whether a pivot should move with the focus ring #

Everything above treats the pivot as fixed and asks where to put it. A panorama head could instead move the pivot as the lens is focused, since the required motion is the extension and the extension is a closed form with one number in it.

Mechanically that is not exotic: a rail that advances the camera by the same amount the lens extends would hold the pivot on the pupil throughout, and a focus scale already carries the information. What is not obvious is whether it is worth building. The gain over a fixed pivot correctly placed is the difference between the walk-limited floor and whatever the focus error contributes at the distances actually used, and a panorama shot at one distance needs no such thing.

The measurement that would settle it takes a scene with subjects at several distances — the near edge of a table at half a metre and a wall at four — and asks what a single panorama costs when the lens is refocused between frames, which is what a photographer with a deep scene does. Each frame then has its own pupil position, so the seam error varies along the sequence rather than being one number, and the question is whether a tracking pivot removes a first-order error or a second-order one.

The short version #

Focusing a lens that moves as one piece carries its entrance pupil forward of the camera body by exactly the extension — the focal length squared over the subject distance less the focal length. For a fifty-millimetre lens that is 2.63 millimetres at a metre and 5.56 at half a metre, and the pupil’s whole walk with field angle over the same traced field is 5.53, so the two are equal at a subject 502 millimetres away and the focus dominates nearer than that.

A panorama head aligned at infinity and used at a metre pivots 2.63 millimetres behind the pupil and leaves 7.46 arcminutes along a sixty-degree seam, against 4.39 for a rig at the paraxial pupil and 1.41 at the best fixed pivot. The error is not monotone in the alignment focus: aligning at four metres leaves 2.21 arcminutes and aligning at about three leaves the full 1.41, because the pivot passes through the best place on its way past it.