Turning and travelling blur different worlds

A frame is an interval held the camera still and let the world move. A point moving across the frame during an exposure draws a straight streak whose length is the focal length times the speed times the exposure, over the depth — so objects at 3 m and 6 m moving at one speed blur by lengths in the ratio 2.000, and no single blur describes the frame.

A photographer following a moving subject does not hold the camera still. The whole point of following is that the subject stays sharp while it moves, and there are two ways to arrange that. The camera can turn to keep the subject centred, the way a spectator follows a runner. Or it can travel alongside at the subject’s speed, the way a camera car follows a cyclist.

Both keep the subject sharp. They do not blur the same things, and the difference is complete: each blurs exactly what the other leaves sharp.

Turning to follow #

A camera that turns about its own centre does not move any ray, so where each still point lands depends only on the direction of the point, not on its distance. A turning frame can be straightened found the same fact for a rolling shutter: a turn moves every point by the same angle at every depth, which is why it can be undone without depth.

For blur, the consequence is that a turn smears the still world uniformly. Following a subject 8 m away at 4 m/s means turning at half a radian a second, and over a 33.3 ms exposure every still object on the line of sight streaks by 12.33 px — at 2 m, at 64 m, and at every depth between — and every still object a third of its depth off to the side by 13.51 px, again at every depth. Near and far are blurred alike; only the direction in the frame changes the length, because a turn moves the edges of a picture further than its middle.

Objects that move with the subject — at its speed, in its direction, but at other depths — are a different matter. The turn is set to keep the subject still, which means it cancels the image motion of something moving at 4 m/s 8 m away. Something moving at 4 m/s 2 m away sweeps across the frame four times as fast as the turn compensates for, and streaks by 37.0 px; something at the same speed 64 m away barely moves across the frame at all, and the turn itself smears it by 10.8 px. The streak is proportional to the difference between one over the object’s depth and one over the subject’s: zero at the subject’s depth and growing on both sides.

So a panning shot has one sharp depth for moving things and no sharp depth for still ones. The background streaks evenly, which is the look a panned photograph of a racing car is prized for: a sharp car against a uniformly streaked world.

Travelling to follow #

A camera that travels alongside at the subject’s speed shares the subject’s motion, and so shares it with everything else moving at that speed.

Everything moving with the subject is now at rest relative to the camera, at every depth, and draws no streak: a rider 2 m away and a lorry 40 m away keeping pace with the subject are exactly as sharp as the subject is. The still world, which the camera is moving past, streaks by the camera’s travel over its depth — the same law as a still camera photographing a moving point, with the roles exchanged. At 2 m the streak is 49.3 px, at 8 m 12.3, at 64 m 1.5.

That is the look of a tracking shot. Near roadside objects are torn into streaks, the distant hills are nearly sharp, and everything travelling with the camera is crisp regardless of how far away it is.

The two, depth by depth #

Set against depth on one plot, the two methods trade places exactly.

Turning, on the line of sight: the still world at 12.33 px everywhere; objects moving with the subject at 37.0, 20.5, 12.3, 4.1, 0.0, 4.1, 6.2, 9.2 and 10.8 px from 2 m to 64 m, with a zero at 8 m.

Travelling: objects moving with the subject at zero everywhere; the still world at 49.3, 32.9, 24.7, 16.4, 12.3, 8.2, 6.2, 3.1 and 1.5 px over the same depths, one over the depth throughout.

Two of those four curves are flat, and they belong to different groups in the two methods. Turning makes still things uniform and moving things depth-graded; travelling makes moving things uniform, at zero, and still things depth-graded.

And the two depth-graded rows are not merely alike in shape. Take the travelling row and subtract 12.3 from every entry: 49.3 becomes 37.0, 32.9 becomes 20.5, 12.3 becomes zero, 1.5 becomes −10.8. Drop the signs and that is the turning row, entry for entry, to 0.04 px. The number subtracted is the travelling streak at the subject’s own depth — and it is also the turning camera’s still-world streak, 12.33 px. The four curves are two curves, each drawn twice.

Following subtracts a motion #

That coincidence has a plain reason, and it is the reason the two methods are each other’s reverse.

Start from a camera that does not follow at all. Still objects draw no streak. Objects moving at 4 m/s draw streaks of the focal length times the distance moved over the depth — the one-over-depth row, 49.3 px at 2 m down to 1.5 px at 64 m, which is exactly the law a frame is an interval measured. A following camera takes that picture and subtracts a motion from every mark in it, chosen so that the subject’s mark stops.

A travelling camera subtracts, from each mark, the motion a still object at that mark’s depth would have — its own depth’s share, one over the depth. Moving objects lose exactly what they had and stop; still objects, which had nothing, gain the whole one-over-depth row with the sign reversed.

A turning camera subtracts the same motion from every mark: the subject’s, 12.33 px. That is what a turn does, as straightening does not move the eye measured — it slides the whole picture by an amount that does not depend on anything in the scene. Moving objects are left with their one-over-depth streak less 12.33, which is zero at the subject’s depth only; still objects gain a uniform 12.33.

So the two methods differ in one choice: whether the motion taken away is a single number or a function of depth. Everything in the four curves follows from that. A dolly zoom is a step and a zoom took a camera motion apart in the same way, into a depth-graded part that belongs to where the camera went and a uniform part that belongs to what the lens did, and found the shot’s whole character in the depth where they cancel.

The symmetry is the same one a scroll is not a panorama drew between an eye that turns and an eye that travels. A panorama keeps one centre and treats every depth alike; a scroll has no centre and sorts everything by depth. A panned exposure is a short panorama of the still world; a tracked exposure is a short scroll of it.

A turn keeps a line of sight still, not a plane #

Turning does not make the whole of the subject’s depth sharp, and the exception is worth measuring because it is where the two methods differ most subtly.

An object moving with the subject at the subject’s depth, but a third of that depth off to the side, still streaks by 1.17 px when the camera turns. A turn moves the edges of a picture further than its middle — by the factor $1 + x^2/f^2$ for a point $x$ pixels from the centre — so a turn that exactly cancels the motion at the centre over-cancels it toward the sides. A panning shot’s sharp region is the line of sight to the subject, spreading slowly into a narrow wedge, and a second car keeping pace beside the first but further across the frame is very slightly blurred.

A travelling camera has no such exception. Everything moving at the subject’s velocity is exactly sharp anywhere in the frame, because the relative motion is zero rather than cancelled.

Reading a photograph backwards #

The difference is visible in a single frame, and it says how the photograph was taken without any information from outside it.

If the still world is streaked evenly — posts near and far, the fence and the stand behind it all smeared by one length — the camera turned. If the still world is streaked by depth — the near railings violently, the distant buildings barely — the camera travelled. The subject is sharp either way, and so the subject says nothing; the background says everything.

The lengths also carry numbers. In a panned photograph the uniform streak is the focal length times the turn over the exposure, so with the exposure known it gives the turn rate, and the turn rate with the subject’s distance gives its speed. In a tracked photograph the still world’s streak at a known depth gives the camera’s speed directly, by the same arithmetic a frame is an interval used, and with the speed known every other still object’s streak gives its distance. Neither reading needs the subject at all.

That second reading is depth is a reciprocal arriving in a single blurred frame. A tracked exposure is, in effect, a stereo pair stretched into a continuum: every still point is seen from every position along the camera’s 13.3 cm of travel during the exposure, and its streak is its disparity over that baseline. It inherits the pair’s weakness with it — the 64 m streak is 1.5 px, and the range a pair cannot see past is reached as soon as a streak’s length is no longer distinguishable from the blur of the lens.

The subject’s own length is the unit #

The pan’s uniform streak has a second meaning, and it makes the shot independent of lens and distance.

The turning camera’s streak is the travelling streak at the subject’s depth, the focal length times the subject’s travel over its distance. The subject’s own image is the focal length times its length over the same distance. Divide one by the other and both the focal length and the distance cancel: the background streak, measured in lengths of the subject’s image, is the distance the subject travelled during the exposure divided by the subject’s length. A 4 m car moving at 4 m/s for 33.3 ms travels 13.3 cm, so the background streaks by a thirtieth of the car’s image length — whether the photographer stood at 8 m with a normal lens or at 80 m with a lens ten times as long, as long as the car fills the frame alike.

That is the panning counterpart of what stepping closer is not zooming found about framing: hold the subject’s drawn size and the change of lens and distance cancels from everything about the subject. It does not cancel from the background’s depth-graded parts, which is where the two setups still differ, but the uniform streak — the thing a panning photographer is actually choosing — is set by the subject’s speed, the exposure and nothing else. A slower car or a shorter exposure gives a proportionally shorter streak, and no choice of position changes it.

A travelling camera has no such unit. Its still-world streaks depend on each object’s depth, so the same exposure following the same car tears a railing 2 m away into a 49.3 px smear and leaves the hills alone, and where the camera stands decides which railing is 2 m away.

Which to choose is a question about the scene #

The four curves turn a stylistic choice into a measurable one, and what decides it is what else is in the frame.

A cyclist in a group, riders at 6 m and 12 m beside one at 8 m, all moving together: panning on the middle rider blurs the other two by 4.1 px each, and a rider at 4 m by 12.3 px; tracking leaves every rider sharp. A subject crossing in front of a still crowd at 2 to 16 m: panning streaks the whole crowd by 12.3 px on the line of sight; tracking streaks the nearest spectators by 49.3 px and those at 16 m by 6.2. The first scene wants a travelling camera and the second has no good answer — which is why a panning photographer standing close to a fence steps back from it, and a tracking vehicle keeps its lens away from roadside posts.

The pivot matters as much as the method. The eye is a place, not a point found that turning about a point behind the lens’s entrance pupil moves the eye a few millimetres; a photographer panning with a camera held at arm’s length moves it by the radius of the arm. That is a small travel added to the turn, and it puts a one-over-depth term back into the still world — negligible on the distant stand and not negligible on a post a metre from the lens.

What neither method does #

Neither makes everything sharp, and the reason is that the scene contains two velocities — the subject’s and the world’s — and a camera can share at most one of them.

A camera that turns shares the subject’s angular motion at one depth, which is shared by nothing else at other depths. A camera that travels shares the subject’s linear motion, which is shared by everything moving with it and by nothing still. The only way to make both the subject and the still world sharp is not to have them moving relative to each other, which is not a photograph of a moving subject.

What the exposure time does is scale everything together. Halving the exposure halves every streak in both figures and changes none of the ratios, which is the same statement a frame is an interval made for a still camera. The choice between turning and travelling is not a way of avoiding blur. It is a choice of which part of the world the blur falls on.

The rows as well as the interval #

The figures here use a global exposure: every pixel integrates over the same 33.3 ms. Every row is a different camera found that most sensors read their rows in sequence, and a turning frame can be straightened found that the turn and the travel separate there too. A turning camera with a rolling shutter shears every depth alike and blurs the still world alike; a travelling one shears and blurs by depth. The two artefacts share their depth structure in each case, which is why a tracking shot of a near roadside has posts that both lean and streak more the nearer they are, and a panning shot has a background streaked and leaning by one amount throughout.

What this does not settle #

Pure motions. The camera either turns about its centre or travels in a straight line. A hand-held camera does some of each, and a camera car following a curve turns as it travels. The combination is the subject of the question below.

Lateral motion only. The subject crosses the frame at a constant depth. A subject approaching or receding draws streaks that fade along their length, as a frame is an interval measured, and following it changes the camera’s focal plane as well as its aim.

Streaks, not appearance. The figures measure how far each object’s image moves during the exposure. How a streak of that length looks depends on the object’s size and contrast and on the sensor, none of which is geometry.

Still open: the sharp distance of a camera that turns as it travels #

A camera that travels at speed $v$ while turning at rate $\omega$ moves each still point’s image by the difference of two terms: the turn’s, which is the same at every depth, and the travel’s, which falls as one over the depth. For one depth the two cancel exactly, at $Z = v/\omega$ — the radius of the circle the camera is moving on.

The question that leaves is whether a camera following a curve has, in its own frame, a sharp distance for the still world: whether a camera car rounding a bend at 10 m/s with a radius of 50 m blurs the still world everywhere except at 50 m, what the streaks look like either side of that distance, and whether the sharp band is a depth or a curve through the scene — since the turn’s contribution grows toward the edges of the frame and the travel’s does not.