A dolly zoom off the axis keeps a line, not a plane

A dolly zoom is a step and a zoom, and they meet at one depth took the shot apart. A camera stepping straight along its line of sight moves every mark outward along the line from the centre of the picture, faster the nearer it is; a zoom moves every mark inward along the same line, by one factor at every depth; and a dolly zoom is both, cancelling exactly on the plane at the subject’s depth. That plane stands still while the nearer world bulges toward the viewer and the further world sinks away.

Every figure in that essay had the camera stepping along its own axis, so the step’s centre and the zoom’s centre coincided at the principal point. A camera on a real dolly track runs along the track, and the track is laid by hand. The question the essay left was what happens when the step’s direction and the camera’s line of sight part: whether a plane of stillness survives, or something less, and how far off the axis a track can run before the held subject visibly moves.

The answer is that the plane does not survive at any angle other than zero, and that what replaces it is a line.

The step radiates from beside the centre #

Take the axial shot’s arrangement — a subject 5 m ahead, a step of 1.5 m, fifty marks at depths from 2.2 m to 40 m — and turn the track 10° from the line of sight in the horizontal plane, leaving the camera pointing where it pointed.

The step alone still moves every mark along a line from a single centre, to $5\times10^{-13}$ degrees, but the centre is no longer the principal point. It is the epipole — the point of the picture the camera is heading toward — 130 px to the left, at the focal length times the tangent of the track’s angle. An epipole in the picture leaves a blind disc found the same point as the centre of the slide every mark makes between two pictures taken along one line. For an axial step it sits at the principal point; turn the track and it moves off by $f\tan\psi$.

The zoom has not changed. Focusing is a zoom and stepping closer is not zooming both rest on the fact that a change of focal length scales the picture about its principal point and nowhere else. So an oblique dolly zoom is a radial expansion about one point laid over a radial contraction about another, and the two no longer share a centre.

The held subject slides #

The hero figure is the whole shot. The step’s flow spreads from the epipole on the left, the zoom’s contracts toward the principal point, and their sum is radial from neither: marks near the subject’s depth drift sideways, those in front of it still stream outward and those behind it still sink inward, but none moves along a line through the centre.

The subject itself moves. Its size is held — the zoom is chosen to hold it — but it slides sideways by

$$\Delta x = \frac{f\,d\sin\psi}{Z_s},$$

where $d$ is the step, $\psi$ the track’s angle and $Z_s$ the subject’s distance. At 10° that is 38.54 px on a picture 690 px wide, to the last digit. The rest of the plane at the subject’s depth slides with it, by the same amount: a plane that stood entirely still in the axial shot now moves as a whole, sideways, away from the epipole. The slider turns the track from 0° to 30°.

The size of the drift is the practical measurement, because a dolly zoom that lets its subject wander is not the shot anyone wanted.

The drift passes one pixel when the track is 0.26° off the axis, for a 1.5 m step on a subject 5 m away with a 740 px focal length. For a subject at 3 m it is 0.15°; at 10 m, 0.52°. A quarter of a degree is a track that drifts 7 mm off line over its 1.5 m. No one lays a track that accurately by eye, so a real dolly zoom on a real track either has its camera turned to follow the subject, or its subject drifts — and at 10° off, by 38.5 px, it drifts obviously.

One point per depth, on one row #

The axial shot had a whole plane of stillness. The oblique one has something much thinner, and it can be found exactly.

A point stays still when the step’s outward push and the zoom’s inward pull cancel at its position. Vertically, the step and the zoom scale a point’s height above the camera’s eye level by factors that cancel only at the subject’s depth — and at that depth the whole plane slides sideways. So off the eye-level row nothing is still. On the row itself there is no vertical motion to cancel, and the horizontal cancels at one column for each depth:

$$x = \frac{f\,t_x\,(Z_s - t_z)}{t_z\,(Z_s - Z)},$$

with $t_x$ and $t_z$ the step’s sideways and forward parts. A search along the row finds each still point where the formula puts it, to a millionth of a pixel. At 10° the still point is 144 px to the left for a point 1.8 m away, 230 px to the left at 3 m, 66 px to the right at 12 m and 6 px to the right at 80 m. As the depth approaches the subject’s, the still point runs off the frame and to infinity: the subject’s plane, the one that stood still in the axial shot, has no still point at all. A metre above eye level nothing is still anywhere; a point there at the column where its depth would be still moves 43.5 px.

So the answer to the question the axial essay left is that there is no plane of stillness and not even a straight line of still points in space. There is a curve at eye height — one still point per depth, hyperbolic in plan — and it lands on a single row of the picture. In the frame, the still points lie along the horizon of the camera’s own height, each depth still at its own column, which is the dolly zoom’s plane of stillness collapsed onto a line.

Why the plane needs the two centres to coincide #

The collapse is not an accident of 10°, and the reason it is total is short.

In the axial shot both motions are radial about one point, so each point’s motion is a single number — a factor on its distance from the centre — and the factor is 1 on the whole plane where the step’s $Z/(Z - d)$ and the zoom’s constant cancel. That cancellation is a condition on depth alone, which is why it holds on a whole plane.

With two centres, each point’s motion is the sum of two vectors pointing along two different lines, and for the sum to vanish the two vectors must be equal, opposite, and along the same line. They lie along the same line only on the line through both centres — the epipole and the principal point — which for a horizontal track is the eye-level row. The condition is then one equation on that row, and it fixes one column per depth. Off the row, two vectors along different lines never cancel, whatever their sizes. The plane of stillness needs the two centres to coincide, and any angle between the track and the line of sight separates them.

Turning to follow keeps the subject and bends its plane #

The practical remedy for the drift is to turn the camera as it moves, keeping the subject on the principal point, and to zoom so that its size is held at its new distance.

That holds the subject exactly: it does not move at all. It does not restore the plane. Other points at the subject’s depth now move by up to 7.8 px across the frame, where in the axial shot they did not move, because the turn sweeps the plane sideways by an amount that changes across it and the zoom and step do not undo that change. Nearer points still stream outward and further ones inward, around the one point that is held.

How the plane moves has a shape. At eye level, a point 2 m to the side of the subject moves 3.8 px on a track 3° off, 12.1 px at 10° and 21.8 px at 20°; a point 1 m to the side moves a quarter of that or a little more, so the motion grows roughly as the square of the distance from the subject — 4.1 to 4.9 times as far at 2 m as at 1 m. And points on both sides move the same way, so the plane is not turning in the picture; it is bending, its middle held at the subject and its sides sagging toward one edge of the frame.

So turning to follow buys one still point and gives up the plane — and, as the next section finds, there is a remedy that buys the plane back. The shot keeps its most noticeable property — the subject holding steady while the world behind it moves — and loses the quieter one, the band of scenery at the subject’s depth that stays locked to the frame. Whether a viewer notices is not a question the geometry answers; what the geometry says is that a subject 2 m wide seen on a 10° track has its edges moving by several pixels relative to its centre.

Shifting the frame keeps the plane #

There is a second remedy, and it does what turning cannot. Without the turn, the subject’s plane did not deform: every point at the subject’s depth slid by the same $f\,d\sin\psi/Z_s$, and in the same direction. A plane that slides rigidly can be put back by sliding the frame after it.

Move the principal point by 38.54 px back toward the epipole — which is what a shift lens does, sliding the lens across the sensor, or what a crop does when the frame is re-centred on the subject after the shot — and every point at the subject’s depth is still again, to $6\times10^{-14}$ px. The axial shot’s whole plane of stillness returns, not only the subject. Nearer points still swell and further ones still sink, though their motions no longer run along lines from one centre, since the step still radiates from the epipole and the zoom from the moved principal point.

The principal point is not the centre measured what a shifted or cropped frame does to a picture’s geometry: a principal point that is not the middle of the frame is an ordinary pinhole picture of a different, off-centre kind. The oblique dolly zoom turns that into a remedy. A camera that can move its principal point — by a shift lens during the shot, or by a crop that slides during the edit — can run on a track that is off the line of sight and still produce the classic shot exactly, with the one condition that the shift grow with the step as $f\,d\sin\psi/Z_s$.

The contrast with turning is exact and has a simple reason. Turning the camera is a rotation, and a rotation moves points at different places in the frame by different amounts: it is a homography of the picture, not a slide. Shifting the principal point is a slide. The oblique dolly zoom’s error on the subject’s plane was a slide, so only a slide undoes it everywhere; a turn can match it at one point, the subject, and misses it by a growing amount elsewhere, which is the bending measured above. A turn of the head is not a step sideways drew the same distinction for a pair of pictures, where the turn leaves no parallax and the step leaves nothing but.

The shot as a camera operator meets it #

Put the three findings together and the oblique dolly zoom is a sharper instrument than the axial one, in both directions.

A track laid along the line of sight to a small fraction of a degree gives the classic shot: a still plane, a subject that holds, near things swelling and far things sinking. A track a few degrees off gives a shot whose subject drifts by tens of pixels unless the camera is turned, and whose still region, with or without the turn, is no longer a plane. A camera turned to follow its subject keeps that subject exactly and bends everything else on its plane by an amount that grows as the square of the distance from it. A camera that shifts its frame instead — with a shift lens, or a crop that slides during the edit — gets the whole plane back, because the error it is correcting was a slide in the first place. Of the two ways an operator has of keeping a drifting subject in place, only the one that does not move the camera’s direction keeps the shot the shot is known for.

Stepping closer is not zooming found that a zoom changes no ratio in a picture and a step changes them all. The oblique shot adds a third ingredient to that pair: the step’s direction. A step along the axis changes ratios radially and symmetrically; a step off it changes them about a different centre, and the difference is visible as soon as the zoom’s centre and the step’s disagree by more than a pixel’s worth of drift.

What this leaves out #

A track that rises or falls. The track here is horizontal, so the epipole lies on the eye-level row and so do the still points. A track that climbs toward the subject puts the epipole above or below the principal point, and the still points then lie on the line through the two centres, tilted in the picture.

A curved track. A dolly on a curve changes its direction of travel during the shot, so the epipole moves during the move, and the still points of the start and the end are on different lines. A camera on a bend is sharp on a circle measured the related case of a camera following a bend during one exposure.

A zoom that moves its own centre. A real zoom lens does not scale the picture exactly about a fixed principal point; its optical centre wanders by some pixels across its range. That wander is a shift of the principal point during the shot, of the same form as the remedy above, so it adds to or cancels part of the drift an off-axis track makes. Its size belongs to a particular lens and was not measured.

What a viewer sees. The drift and the bending are measured in pixels. Whether a subject sliding by a pixel or an edge bending by several is noticed on a screen is a question about perception, and not one the geometry settles.

Still open: a dolly zoom along a climbing track #

The track here was level, so its epipole shared a row with the principal point and the still points lay along that row. A track that rises toward the subject — a crane move, or a dolly on a ramp — puts the epipole directly above or below the centre, and the still points should then lie along the vertical through the centre, one per depth, while a track that both turns and rises puts them on a slanted line through the two centres.

The measurement that settles it tilts the track by a stated elevation as well as turning it, finds the still points by search, and asks whether they lie on the line through the epipole and the principal point for every combination of the two angles — and, since a track that rises changes the camera’s height above the ground during the shot, whether the horizon of the ground itself moves in the frame in a way that gives the move away independently of any still point.