A dolly zoom is a step and a zoom, and they meet at one depth

Stepping closer is not zooming proved that a change of focal length leaves every ratio in a picture alone, that a step changes them, and that holding a subject’s drawn size while stepping in — the dolly zoom — shrinks the background by a factor it wrote in closed form. The essay measured the shot through two numbers, the subject’s size and the background’s.

A dolly zoom is watched, not measured at two points, and what a viewer sees is motion everywhere in the frame. So the question that follows is what the whole picture does: where each mark goes as the camera steps and the lens shortens, and why the effect reads the way it does.

The answer turns out to be exactly two motions laid over each other, each simple, and a depth at which they cancel.

Every mark moves along a line from the centre #

Fifty points spread through the frame at depths from 2.2 m to 40 m, the camera stepping 1.5 m toward a subject 5 m away, the focal length shortening in step to hold the subject’s size. Each mark’s motion is an arrow in the picture, and every arrow lies along the line from the principal point through the mark, to $10^{-13}$ degrees. Nothing moves sideways, nothing rotates, nothing shears.

Along those lines, marks nearer than the subject move outward — the nearest, at 2.2 m, to 120 per cent further from the centre than they were. Marks further than the subject move inward, the most distant toward 70 per cent of their distance. Marks on the plane 5 m away do not move at all.

That is the whole motion field, and it is simple enough to be described by one number per depth. The number is worth taking apart, because each of its two factors is a familiar operation.

The step alone #

Keep the focal length and only step 1.5 m forward.

Every mark still moves along the line from the centre, and every one moves outward. The factor is $Z/(Z - d)$ for a point at depth $Z$ and a step $d$: 214 per cent further out at 2.2 m, 43 per cent at 5 m, 3.9 per cent at 40 m. Near things rush out of the frame and far things barely move.

The reason the arrows radiate from the centre is that a camera stepping along its own axis is heading straight at the image of its own direction of travel, and an epipole in the picture leaves a blind disc found what that means: the image of the other eye sits at the centre of the frame, and every mark slides directly away from it between the two pictures, faster the nearer it is. The step’s flow is that slide. Its centre is the epipole, which for a step along the axis is the principal point.

The zoom alone #

Keep the camera where it is and only shorten the focal length by the factor the dolly zoom uses.

Every mark moves inward along the line from the principal point, and every one by the same factor, 0.70, at every depth — the measured factors run from 0.700000 to 0.700000. A zoom is a scaling of the picture about its principal point and nothing else, which is the invariance stepping closer is not zooming proved: no ratio in the picture changes, because every distance from the centre is multiplied by one number.

Focusing is a zoom found the same operation hiding inside a focus ring: a lens moving away from the sensor to focus close lengthens the principal distance and scales the picture about its centre, exactly as a zoom would. For the dolly zoom it does not matter which mechanism shortens the principal distance; only the factor does.

The composition #

A dolly zoom is the step and the zoom applied together, and since both scale distances from the same centre, their factors multiply.

The zoom’s factor is fixed by the requirement that the subject’s size not change: the step multiplies the subject’s distance from the centre by $Z_s/(Z_s - d)$, so the zoom must multiply it by $(Z_s - d)/Z_s$. The product, for a point at depth $Z$, is

$$\frac{Z_s - d}{Z_s}\cdot\frac{Z}{Z - d},$$

and the fifty drawn marks match it to $9 \times 10^{-16}$. It is 1.750 at 2.5 m, exactly 1 at the subject’s 5 m, 0.800 at 12 m, and tends to 0.700 as the depth grows, because the step’s factor tends to 1 for things infinitely far away and only the zoom’s remains.

That is where the reversal comes from, and it is exactly one depth. Nearer than the subject, the step’s outward push is stronger than the zoom’s inward pull; further, the zoom wins; at the subject’s depth they cancel identically, and every point on that plane — not just the subject — stays where it was. The held plane is a whole plane of stillness through the scene.

It is a plane and not a sphere, which is easy to get wrong when picturing the shot. The factor depends on a point’s depth along the camera’s axis and on nothing else — not on how far it is to the side, and so not on its straight-line distance from the lens. A point 30° off the axis at the subject’s depth is 5.77 m from the camera, not 5, and it stays exactly as still as the subject does. A wall parallel to the picture at 5 m holds still from edge to edge; a curved backdrop at a constant 5 m distance would not, its edges being nearer than the plane and moving outward.

Where the reversal sits #

The held depth is a choice, and each choice moves the point of stillness and the limit together.

Holding a subject at 3 m, the same 1.5 m step needs the focal length halved, so distant things shrink toward half their distance from the centre. Holding at 5 m, toward 0.70. Holding at 10 m, toward 0.85. The nearer the subject, the harder the same step pulls the background in, because the step is a larger fraction of the subject’s distance and the zoom has to compensate for more.

That restates the rule stepping closer is not zooming derived for the background’s size — everything depends on the step in units of the subject’s distance — as a statement about the whole field. The curve for each held depth has the same shape, shifted and squeezed, and its limit is $1 - d/Z_s$.

The flow is a depth map #

The factor is one number per mark, and it can be read backwards.

Write the step as a fraction of the subject’s distance, $\delta = d/Z_s$, and each mark’s depth in the same unit, $z = Z/Z_s$. The factor is then $k = (1-\delta)/(1 - \delta/z)$, and solving for the depth gives

$$\frac{1}{z} = \frac{1}{\delta}\left(1 - \frac{1-\delta}{k}\right).$$

Everything on the right is in the picture. A mark’s factor is how far it ends up from the principal point over how far it started, and $1-\delta$ is the factor the most distant marks approach, 0.70 here, so the step is read off the flow’s own limit. Put in the factors the figure measured and the depths come back: 1.750 gives 2.5 m, 0.800 gives 12 m, and 1 gives the subject’s plane.

What comes back is inverse depth, linear in one over the factor, and that is depth is a reciprocal arriving through a zoom lens. It inherits a stereo pair’s precision with it. Take a mark 200 px from the centre, read so that its factor is known to 0.00125 — a quarter of a pixel in 200. Propagated through the formula, that places a point at 5 m to 1.5 cm, at 8 m to 5 cm, at 12 m to 13 cm, at 20 m to 41 cm and at 40 m to 1.8 m: from 0.3 per cent of the depth at the subject to 4.4 per cent eight times further out. An error growing as the square of the depth is the signature the range a pair cannot see past measured for two cameras side by side, and it is here for the same reason. A dolly zoom is a stereo pair along the axis with a change of scale laid over it, and the change of scale is uniform, so it carries no depth and costs none.

And what comes back has no size. Every quantity in the formula is a ratio to the subject’s distance, so the flow gives the scene’s depths as multiples of the subject’s and the step as a fraction of it, and no reading of the flow says whether the subject stood 5 m away or 50. That is the one number two views give shape and no size found missing from any pair of pictures, and a dolly zoom, for all its strangeness, leaves the same number undetermined.

How far the shot can go #

A longer step makes a stronger effect, and two things run out.

The first is the lens. The zoom’s factor is $1-\delta$, so a camera starting on a 50 mm lens and holding a subject 5 m away needs 35 mm for a 1.5 m step, 25 mm for 2.5 m and 10 mm for 4 m. The background’s limit is the same number: stepping 4 m of the 5 draws the far distance in toward a fifth of its distance from the centre. A zoom lens’s range is therefore the largest fraction of the subject’s distance the shot can cover — a threefold zoom allows two thirds of the way in, a tenfold zoom nine tenths.

The second is the foreground. The step’s factor $Z/(Z-d)$ grows without limit as a point’s depth approaches the step, because depth $d$ is where the camera ends up. With a 1.5 m step, the marks at 2.2 m already end 120 per cent further out; a mark at 1.6 m would end eleven times as far from the centre as it began, off any frame. So the nearest thing in the picture bounds the step: anything nearer than the distance the camera moves is not stretched but passed. The geometry asks for the ground in front of the subject to be empty to the depth the camera travels, which is why the shot is made down corridors, across open floors and along empty roads.

Focus does not drift #

A real lens refocuses as the subject’s distance falls from 5 m to 3.5 m, and focusing is a zoom found that refocusing changes the principal distance — which would add a small zoom of its own and make the held subject breathe. It does not, and the reason is exact rather than small.

A thin lens of focal length $f$ focused at distance $u$ has principal distance $v = f/(1 - f/u)$. Holding the subject’s size makes the focal length shrink in proportion to the subject’s distance, so $f/u$ stays fixed: 50 mm at 5 m and 35 mm at 3.5 m are both a hundredth. The refocusing factor $1/(1 - f/u)$ is then the same at every instant of the shot, and the principal distance scales exactly as the focal length does. Whether the ratio is set on the focal length or on the principal distance, it is the same ratio, and the subject holds.

The centre does not move #

One consequence of the flow’s form is easy to miss and explains how the shot looks.

Every mark moves by its factor times its distance from the principal point. A mark near the centre is a small distance from it, so it moves a small amount whatever its depth; a mark at the edge of the frame moves a lot. The dolly zoom’s motion is largest at the edges of the picture and vanishes at its centre, for every depth.

That is the blind disc of a forward step arriving in a different role. An epipole in the picture leaves a blind disc found that near the epipole a forward step’s slide is too small to recover depth from. The dolly zoom uses exactly that property on purpose: the subject sits at the centre, where no step moves anything much, and the zoom holds it still exactly; the background around it, spread across the rest of the frame, is where the step’s depth-dependence shows.

It also says why the effect is usually framed as it is. A subject placed off to the side of the frame while the camera dollies along its axis would still keep its size — the factor at its depth is 1 wherever it is — but it would not keep its place relative to a background that moves more near the frame’s edges. The centred subject is not only the conventional composition; it is the composition in which the subject is both held in size by the zoom and held in position by the geometry.

The motion a viewer has to explain #

Stepping closer is not zooming argued that the shot unsettles because it holds one depth cue fixed while another changes, and was careful to say it could not establish why a viewer reacts to that. The flow field sharpens what is being held fixed and what is changing, without settling the reaction either.

In an ordinary forward step, every mark moves outward; the flow is a single expansion from the direction of travel, and it is the flow a moving observer normally sees. In an ordinary zoom, every mark moves inward by one factor; the flow is a uniform contraction, and it is the flow of an image being magnified. A dolly zoom is neither. It has an expansion and a contraction in one picture, separated by a plane of stillness through the scene, and there is no single motion of a camera with a fixed lens that produces that field. A picture with no size–distance signal found that removing a cue leaves a picture consistent with many scenes; this flow is consistent with no motion of a rigid camera at all, and that is the precise sense in which the shot’s evidence disagrees with itself.

What a visual system does with evidence of that kind is outside what this measurement can say.

What this does not settle #

A step along the axis. The step and the zoom share a centre only because the camera steps along its own optical axis, putting the epipole at the principal point. A dolly along any other line has its expansion centred elsewhere, and the composition would no longer be radial about one point. That case was not drawn.

Small points, not surfaces. The flow is measured for isolated points. A real background is a surface whose parts at different depths move differently, so it is stretched as well as moved; that stretching follows from the same factor and was not drawn.

The lens as a pinhole. The focal length is changed without moving the centre of projection. The entrance pupil walks with the angle found that a real lens’s centre moves as it zooms and focuses, which adds a small step of its own to every dolly zoom.

Still open: a dolly zoom along a line that is not the axis #

Every figure here has the camera stepping straight along its line of sight, so that the step’s centre of expansion coincides with the zoom’s centre of scaling. A camera on a track that runs at an angle to where it points has its expansion centred on a point off to one side — the epipole moves off the principal point — while the zoom still scales about the principal point.

The question that leaves is what the composed flow does then: whether there is still a plane of stillness, or only a line of still points where the two centres’ fields happen to cancel; how far off the axis a track can run before the held subject visibly drifts across the frame; and whether the shot survives as an effect at all once its two motions no longer share a centre.