A camera on a bend is sharp on a circle

Turning and travelling blur different worlds followed a moving subject two ways and found that each blurs exactly what the other leaves sharp. A camera turning on the spot to keep the subject centred smears every still thing by the same amount at every depth, 12.33 px on the line of sight over a thirtieth of a second; a camera travelling alongside smears the still world as one over its depth, 49.3 px at 2 m and 1.5 at 64 m.

A camera that turns as it travels does both at once. The turn’s image motion is the same at every depth and the travel’s falls as one over the depth, and the two point in opposite directions when the camera turns toward the side it is looking at — so for one depth they cancel exactly, at the speed divided by the turning rate. That is the radius of the circle the camera is moving on. The earlier essay ended by asking whether a camera car rounding a bend therefore has a sharp distance for the still world, what the streaks look like either side of it, and whether the sharp band is a depth or a curve through the scene.

It has one, and it is not a depth.

The bend as one rigid turn #

The measurement is set up the way the question put it. A camera car rounds a bend of 50 m radius at 10 m/s, turning with the road at a fifth of a radian a second, and its camera is aimed at the bend’s centre, looking into the curve. Each frame is a 33.3 ms exposure with a 50° field, and every still point in view draws a streak whose length is how far its image moves while the shutter is open.

Before measuring, the geometry can be said in one sentence, and it is worth saying because it decides everything that follows. A camera moving round a circle and turning with it is a rigid body rotating about the circle’s centre. In the camera’s own frame, therefore, the whole still world rotates about that centre — every still point sweeps a small arc about it during the exposure — and a point’s image stays still exactly when that arc runs along the point’s own line of sight, toward or away from the eye.

A point’s arc is perpendicular to the line from the bend’s centre to the point. It runs along the line of sight when the angle at the point, between the direction to the centre and the direction to the eye, is a right angle — and the points that see a fixed segment at a right angle lie on the circle with that segment as its diameter. So the still world is sharp, at eye height, on the circle whose diameter joins the camera to the bend’s centre.

The plan bears it out. Sampling the field every two metres in range, the points whose streak is under a pixel lie in a band hugging that circle: 478 of the 1,715 sampled. Nearer than the circle they streak by up to tens of pixels; beyond it they streak by a few pixels, the other way.

Straight into the bend #

Along the camera’s own axis the circle is met once, at its far end, which is the bend’s centre fifty metres away.

Travelling alone would streak a point on the axis by the focal length times the distance travelled over the range — 61.7 px at 4 m, falling as one over the range. Turning alone would streak everything by 4.93 px. Doing both, the two subtract: 56.7 px at 4 m, nothing at 50 m, and back toward 4.93 px far away, 4.32 px at 400 m. The streak is exactly the focal length times the exposure times the speed times the difference between one over the range and one over the radius, so it is under a pixel from 41.6 m to 62.7 m.

The band is lopsided for the reason every band in the depth field is lopsided. Depth is a reciprocal found a fixed error in a reading mapping to an interval of depth that is not centred on the answer, and a fixed tolerance in one over the range does the same here: 8.4 m in front of the centre and 12.7 m behind it.

And the streak changes direction across the centre. Nearer than fifty metres the camera’s travel carries points past faster than its turn sweeps them back, and they streak one way; beyond fifty metres the turn wins and they streak the other. That is the visible signature of a turning-and-travelling shot, and it is worth seeing in a frame.

The posts at 8 m streak by up to 27 px toward one side; those at 200 m by about 4 px toward the other. At 50 m the post on the axis is a point — 0.00 px — and those 10 and 20 degrees off the axis streak by 0.08 and 0.34 px. They are close to sharp and not exactly sharp, which is the circle again: at 50 m they are behind it.

The sharp place is a circle, not a distance #

The circle can be tested directly. Along each bearing off the axis, search for the range at which a still point’s streak vanishes.

The search finds 50.0 m on the axis, 47.0 m at 20 degrees, 43.4 at 30 and 35.5 m at 45 degrees, against the circle’s 50 times the cosine of the bearing, and the two agree to 0.33 per cent. The difference is the exposure’s own. In 33.3 ms the camera turns 0.38 of a degree, so the exact set of points whose two images coincide is a circle through the camera and the centre tilted by half that turn, rather than the one with them as its diameter; the circle in the geometry is the instantaneous version, and the measurement is of a real exposure.

So the answer to “is the sharp band a depth or a curve” is a curve, and a particular one: at the edge of a wide field it is far nearer than at the middle. A photographer who focuses a lens on the bend’s centre and hopes the still world at that distance will be sharp across the frame gets the middle and not the edges, because the sharp place at the edges is fourteen metres nearer.

The circle has turned up before, between two eyes rather than two moments. The depth a pair calls zero is about the Vieth–Müller circle, the set of points two verged eyes see with no disparity, which is a circle through both eyes for the same inscribed-angle reason. There the circle is where two views of one moment agree; here it is where one view at two moments agrees. The camera car’s frame is a stereo pair taken a thirtieth of a second apart, from two points on the bend, and its sharp place is that pair’s horopter.

Two motions that cancel, once before #

This is the second time the sensor field has found a sharp place made by two motions cancelling rather than by a lens. A dolly zoom is a step and a zoom, and they meet at one depth found that stepping toward a subject while shortening the lens to hold its size moves every mark along a line from the centre of the picture — outward for anything nearer than the subject, inward for anything beyond — and leaves one depth still. There the two motions were an expansion and a scaling about the same point; here they are a sideways travel and a turn about a vertical axis. Both times the still place is where a motion that depends on depth meets one that does not, and both times it separates a near world moving one way from a far world moving the other.

The difference is in the shape of the still place. The dolly zoom’s two motions share a centre, so its still place is a plane square to the axis. The bend’s travel and turn do not share a centre — the travel is along the road, the turn is about the bend’s middle — and that is what bends the still place into a circle. A camera that travelled toward the bend’s centre instead of round it, zooming as it went, would be the dolly zoom again.

Only one line stays sharp to the sky #

Everything above is at eye height. The circle is a statement about the ground plane, and what happens off it is different.

The bend’s centre stays sharp at every height — its streak twenty metres up is 6 × 10⁻¹⁴ px — because it lies on the axis the whole scene rotates about, and a point on the axis of a rotation does not move at all. Everywhere else on the circle, a raised point moves along a horizontal arc while its line of sight tilts upward, so its motion is no longer along the ray, and its streak grows in proportion to its height: 20 m up it is 0.36 px at 10 degrees off the axis, 0.81 px at 20 and 1.52 px at 30.

So the sharp place, fully stated, is a circle on the ground and one vertical line above it. A tree standing at the bend’s centre is sharp from root to crown. A tree on the circle thirty degrees off the axis is sharp at its foot and streaked at its top. The frame shows a single sharp column rising out of a sharp arc, which no still camera and no straight-running one could produce.

Only a camera looking into the bend has one #

The last question is the direction the camera looks. A camera car filming a bend can point into it, along the road, or out of it, and the circle does not move with the camera.

Turn the camera from the centre toward the road ahead and its axis meets the circle nearer the camera: 50.0 m straight at the centre, 43.4 m at 30 degrees of turn, 25.1 m at 60 and 4.5 m at 85, fifty metres times the cosine of the turn. At 90 degrees — looking straight along the road — the axis is tangent to the circle at the camera itself, and nothing ahead is sharp: the least streak along the axis is 4.93 px, which is the turn’s own streak far away. Aimed out of the bend, the circle is behind the camera and the least streak is 5.15 px.

That is the practical answer to the camera car. A forward-facing camera on a bend has no sharp distance for the still world, which streaks everywhere by at least what the turn alone would give. A side-facing camera on the inside of the bend has one, and it is the bend’s centre. A side-facing camera on the outside has none. The same car, the same speed and the same bend give three different pictures depending on which window the camera looks out of.

What a photographer can do with it #

The earlier essay’s two ways of following a subject each had a signature: turning blurs the still world evenly, travelling blurs it by depth. The bend adds a third signature, and it is a usable one.

The centre of a bend is a free sharp point. Anything standing at the centre of the curve the camera is following is sharp without any tracking, and sharp to its full height. A photographer shooting from a car rounding a roundabout, aimed at its middle, gets the statue in the middle sharp at shutter speeds that streak everything else.

The sharp band is shallow and lopsided. At these numbers a pixel’s tolerance holds from 41.6 m to 62.7 m on the axis, and nearer at the edges of the frame. That is a depth of field set by motion rather than by focus, and it is in the same place whatever the aperture — the sharp band is a decision is about the other kind, the one a lens sets, and the two do not have to coincide.

And the speed does not move it. The sharp range is the speed over the turning rate, which for a car on a road is the bend’s radius whatever the speed. Driving faster makes everything else streak more and leaves the sharp circle where it was.

Where this reading stops #

The camera turns exactly with the road. Every figure has the camera fixed to the car and the car following the bend exactly. A camera on a gimbal that turns faster or slower than the road has a sharp range at its own speed over its own turning rate, which is a different circle; nothing here measures a camera that turns independently.

A global shutter. Each exposure here is taken all at once. A rolling shutter reads the rows at different moments, and every row is a different camera is about what that does to a frame from a moving camera; on a bend, each row would presumably have its own sharp circle, and how far they differ was not measured.

The ground is flat and the road level. A bend that climbs or banks adds a rotation about another axis, and the rigid-turn picture then has an axis that is not vertical. The sharp line would tilt with it.

And streak length is not all of blur. A streak is how far a point’s image moves; what a picture shows also depends on the shutter’s timing and the lens’s own blur. A frame is an interval set up streak length as the measure, and it is the right one for comparing where a frame is sharp, not for saying how sharp it looks.

The bend, answered #

A camera that turns as it travels is a rigid body rotating about the centre of its path, so the still world rotates about that centre in the camera’s frame, and a still point’s image stands still where its motion runs along its line of sight. At eye height that is the circle whose diameter joins the camera to the centre; off the ground it is the vertical line through the centre alone.

For a camera car on a 50 m bend at 10 m/s, aimed into the bend, the streak on the axis is f·T·v·|1/Z − 1/R|: 56.7 px at 4 m, nothing at 50 m, under a pixel from 41.6 to 62.7 m, and changing direction across the centre. Off the axis the sharp range is 50 m times the cosine of the bearing, to 0.33 per cent — 35.5 m at 45 degrees. The centre is sharp at any height; a point on the circle 30 degrees off the axis streaks 1.52 px twenty metres up. Turned toward the road the sharp range falls as the cosine of the turn, and aimed along the road or out of the bend the camera has no sharp distance at all.

Still open: what a rolling shutter does to the sharp circle #

A rolling shutter reads the frame one row at a time, so each row is a camera at a slightly different point on the bend, turned by a slightly different amount. A turning frame can be straightened found that a pure turn read row by row can be undone exactly, because a turn moves every point by an amount that does not depend on depth, and that a travel cannot, because it does.

On a bend the camera does both, and the two cancel on the circle. That suggests a question with a number in it: whether the points on the sharp circle are also the points a rolling-shutter frame puts in the right place — whether the circle that is free of streaks is also free of shear. If it is, the bend’s centre is not only a sharp point but an undistorted one, and a frame could be straightened about it without knowing any depth. If it is not, the row-by-row readout moves the sharp place, and the measurement that settles it is how far the circle a rolling frame is sharp on sits from the global-shutter one, against the readout time.