The rows read where the camera rides on the body, and a camera behind the centre hides the pitch

A body that bounces leaves a second swelling in the rows gave a vehicle’s body two modes on its springs, a pitch at 1.4 hertz and a bounce at 1.0, and found that a two-slit scroll camera’s rows along a rough road carry both: the pitch by turning the camera, the bounce by lifting it through the height the eye climbs between the slits’ two moments. One 800-metre pass read both frequencies, the pitch to ±0.02 hertz and the bounce to ±0.03. The fit was told one thing it did not read, how far ahead of the body’s centre the camera stands, and a fit not told it read a 1.4 hertz pitch as 1.5.

That distance, the camera’s lever arm, enters the rows in a particular way. A camera out on the arm rises and falls with the pitch, by the arm times the pitch angle, so the arm adds a copy of the pitch’s own motion to the camera’s lift. The essay ended by asking whether that copy draws a shape plain enough for one pass to read the arm as a fifth number beside the two frequencies and two dampings, what reading it costs them, and whether a body that does not bounce, whose lift then carries nothing but the arm, reads it better.

It does read it, at almost no cost, and the body without a bounce reads it twice as well near the centre and no better far from it. The measurement also found something the question did not ask about. A camera behind the centre is the hardest place of all, because there the arm’s lift works against the pitch’s turn.

One pass reads where the camera stands #

The vehicle is the earlier essays’: wheels 2.7 metres apart, the scroll camera looking sideways at a line of posts twelve metres out every quarter metre, rows read to a tenth of a pixel, driven at ten metres a second over 800 metres of a good road. The body pitches at 1.4 hertz and, in the first case, bounces at 1.0, both damped 0.3. The camera stands anywhere from a metre behind the body’s centre, as on a van’s rear door, to 2.5 metres ahead of it, as on a long bonnet. Each of twenty roads is driven once and read by the two-mode fit with the arm left free.

The arm comes back. A camera 1.5 metres ahead of a body that pitches and bounces is read at 1.48 metres, give or take 0.15; over the centre, at 0.00 ± 0.11; 2.5 metres ahead, at 2.47 ± 0.20; a metre behind, at −0.87 ± 0.17. Every reading ahead of the centre lies within a tenth of its own spread of the truth, and the one behind within one spread.

A body that does not bounce is read better near the centre and no better further out. Over the centre its arm comes back as 0.01 ± 0.05, twice as tight as the bouncing body’s; half a metre ahead, ±0.06 against ±0.11; a metre, ±0.09 against ±0.13. By two metres the two are alike, ±0.17 each, and at 2.5 metres and a metre behind, the stiff body is slightly worse. The earlier essay’s guess was that the arm would be read better where the lift channel has nothing else to say. That holds only near the centre, where the arm’s share of the lift is small and a bounce mixed into the same channel hides it. Far out, the arm’s copy of the pitch dominates the lift whether the body bounces or not, and the spread is the arm’s own.

The arm is a shape in the spectrum, not a level #

Why the arm can be read at all is a question about shape. The two-mode fit already carries the road’s level, a single multiplier on the whole spectrum that it fits along with the modes; anything the arm did evenly across every wavenumber would be absorbed into that level and never seen. The figure shows that it does nothing evenly.

A camera a metre ahead of the centre multiplies the rows’ answer by 5.5 near 0.125 cycles a metre, just short of where the pitch rings at this speed, and by 1.06 at the short end of the band. At 2.5 metres ahead, by 21.7 under the pitch and 1.37 at the short end. The extra lift is a copy of the pitch, so it swells where the pitch swells and fades where the pitch fades, and the swelling’s height against its shoulders is what the fit reads the arm from. One pass reads the road’s slope as well as its spring read the road’s spectral slope the same way: a shape the spring’s swelling does not share.

Behind the centre the copy has the opposite sign. A camera a metre behind the centre sinks when the nose rises, and the rows answer that sinking against the turn rather than with it. With the bounce in the body, the answer falls to 0.30 of the centred camera’s near 0.18 cycles a metre. Without the bounce, the cancellation near the pitch is all but complete: the rows’ answer at 0.13 cycles a metre is a thousandth of what it is with the camera at the centre. A camera a metre behind the centre of a stiff body hardly sees the pitch where it rings, and has to read it from its flanks.

A camera taken to be at the centre reads the pitch wrong both ways #

The earlier essay found a fit that does not know the arm reading a 1.4 hertz pitch as 1.5 from 1.5 metres ahead. The figure extends that to every place on the body, and the bias turns out to run both ways.

Ahead of the centre a fit that puts the camera there reads the pitch high: for the bouncing body 1.44 hertz from half a metre, 1.49 from 1.5 metres and 1.54 from 2.5; for the stiff body, 1.50, 1.57 and 1.59. The stiff body is more biased because its rows have no bounce mode for the misplaced lift to escape into: the whole of the arm’s copy is charged to the pitch’s shape. Behind the centre the bias reverses and grows: a metre back, the bouncing body’s pitch is read at 1.25 hertz and the stiff body’s at 1.80, nearly a third more than it is. With the pitch’s own swelling almost cancelled, the fit that does not know why chooses whatever frequency best explains the flanks it can see.

With the arm free the bias is gone at every place. The bouncing body’s pitch is read at 1.38 to 1.40 hertz throughout. The stiff body’s is read at 1.39 to 1.40 everywhere but a metre behind the centre, where it comes back as 1.44 ± 0.21. The reading is no longer wrong there, but it is ten times wider than anywhere else, because with the pitch’s swelling cancelled the arm and the pitch’s frequency trade against each other over the same faint flanks. That place on the body is the one to avoid, whether or not the arm is fitted.

What the fifth number costs #

A fit with one more unknown usually reads the others worse, and how much worse is the price of not measuring the arm with a tape. For the pitch the price is nothing. With the camera half a metre ahead, the pitch is read to ±0.017 hertz with the arm known and ±0.017 with it read; at 1.5 metres, ±0.018 both ways; at 2.5 metres, ±0.021 and ±0.019. The pitch’s swelling is the stronger in the rows wherever the camera stands ahead of the centre, and its place along the road’s wavenumbers does not move with the arm, which only changes its height. Nothing about where the swelling sits trades with how tall it is.

The bounce pays. Its frequency is read to ±0.017 hertz half a metre ahead with the arm known and ±0.023 with it read; at 1.5 metres, ±0.026 and ±0.035; at 2.5, ±0.038 and ±0.046. That is a quarter to a third more spread, and the reason is the same channel that made the arm readable. The bounce is read from the lift the pitch does not explain, and the arm moves some of the lift from one to the other. Fitting the arm means letting the fit decide how much of the lift is the pitch’s copy and how much is the bounce’s own, and that decision costs the bounce a little of its precision.

A third of the bounce’s spread is the full price of the fifth number. Against it stands the bias of getting the arm wrong: the previous figure’s tenth of a hertz in the pitch from a camera taken to be at the centre when it stands 1.5 metres ahead. A reader who knows the arm to a few centimetres should give it; a reader who does not loses less by fitting it than by guessing it.

The arm narrows with the road read #

Every reading from the rows’ spectrum has narrowed with the length of road driven, because the spectrum is averaged over fifty-metre windows and a longer pass gives it more of them. The arm is no exception. From 200 metres of road it is read to ±0.36 metres; from 400, ±0.21; from 800, ±0.15; from 1600, ±0.09. Each doubling narrows it by about the square root of two, as the dashed line predicts, a little faster from 200 to 400 metres and a little slower beyond.

The shortest pass is also biased. From 200 metres the arm is read at 1.34 metres on average, against the true 1.5, and the bounce at 1.05 ± 0.20 hertz instead of 1.00. Four windows of road are too few to draw the shoulders of the pitch’s swelling clearly, and the fit pulls the arm towards the middle of the range it searched. From 400 metres on, the mean is right to a couple of centimetres. A reader who wants the arm to a decimetre needs about a kilometre and a half of road, at this speed and reading precision, in one pass or several.

The earlier essays found the same length doing the same work for the spring’s frequency, its damping and the road’s slope: the pass is the instrument, and its length is its aperture. A longer pass is free on a road that goes on, which is why the arm can be fitted rather than measured.

How close the bounce can come #

The arm is the share of the camera’s lift that moves in step with the pitch rather than with the bounce, and reading a share needs the two motions to be told apart. The earlier essay found the bounce’s frequency read well while it lies a few tenths of a hertz from the pitch and badly once it comes within a tenth; the arm should follow.

It does. With the bounce at 0.8 hertz, six tenths below the pitch, the arm is read to ±0.10 metres; at 1.0, ±0.15; at 1.2, ±0.19; at 1.3, ±0.19. The mean stays right throughout, between 1.48 and 1.50. The bounce’s own frequency, with the arm free, is read to ±0.03 hertz at 0.8 and 1.0, ±0.10 at 1.2, and ±0.20 at 1.3, where it is barely a reading. The pitch is read to ±0.03 hertz at worst. When the two modes ring over the same stretch of the road’s wavenumbers, the two lifts are one swelling and the arm’s share in it is hard to see, though the swelling’s height, which the pitch owns, is not.

So the arm is read best on a body whose bounce lies well below its pitch, which is the usual order for a car, and its spread doubles as they approach, without bias.

What a survey camera should do with its own mounting #

A road survey that scrolls the roadside does not choose where its camera goes for the sake of the rows. It goes on a roof bar, a bumper or a tailgate, wherever the view is clear, and the distance from the body’s centre is whatever that place happens to give. A scroll is a camera that moves made the motion the instrument; these measurements say the mounting is part of the instrument too, and that the instrument can measure it.

Three practical rules come out of the figures. Mount ahead of the centre where possible: the arm then adds to the pitch’s swelling rather than cancelling it, and the pitch is read to two hundredths of a hertz whether the arm is known or fitted. Avoid a metre or so behind the centre, the one place where the pitch hides from the rows and every reading of it widens. And drive a kilometre or more of ordinary road before trusting the arm the rows give; four hundred metres, the length a rough road rings the suspension well enough to read found enough for one spring, reads the arm only to two decimetres.

None of this touches the ranging the scroll was built for. A scroll through two slits ranges in a straight line read depth from the gap between a point’s two drawings, and every essay since has been about what disturbs that gap. The body’s springs disturb it, the camera’s place on the body decides how, and the centre a scroll does not have is, in this sense, joined by a second missing centre: the point on the body the camera’s motion is measured from, whose distance from the camera the rows now read.

Why the place on the body matters #

Put together, the rows of a scroll camera driven once over a kilometre of ordinary road read where the camera is mounted on the body, to a decimetre or two, alongside the two frequencies and two dampings of the body’s springs; and reading it costs the pitch nothing and the bounce a quarter to a third of its spread. A reader who takes the camera to be at the centre, as every essay before the last one did, reads the pitch high from ahead of the centre and wrong in either direction from behind it.

A scroll camera rings with its vehicle’s suspension was the first of these essays to put a spring under the camera, and it put the camera at the centre of the body by assumption. A vehicle’s pitch lags the road by its wheelbase before it gave the body no springs at all. The assumption was harmless while the body was rigid, because a rigid body’s pitch is the chord between its wheels wherever the camera sits on it. A body on springs rotates about some point, and the camera’s place relative to that point decides whether its lift adds to its turn or subtracts from it.

The result that most changes practice is the one behind the centre. A camera a metre behind the centre of a car that does not bounce is nearly blind to the pitch where the pitch rings, and the fit, told or not, reads the frequency there worst. A scroll of a climbing road measures its grade began this line of argument by finding the rows’ answer to a change of height; here that weaker answer, set against the stronger answer to a turn, can erase it.

What the reading takes from outside the rows #

The body pitches about its centre. The arm is measured from the point the pitch turns about, which the model puts at the body’s centre between the axles. A real body pitches about a point set by its spring rates front and rear, and what the rows read is the distance from that point, wherever it is.

The two modes are uncoupled. Real pitch and bounce mix when the front and rear springs differ, and each mode then moves the camera in both ways at once. The fit would read an arm that is partly the coupling.

The road’s slope is known. The arm and the road’s spectral slope both change how the spectrum leans across the pitch’s swelling. This essay held the slope at the survey standard; the earlier essay read it free, and freeing both would trade them.

One reading precision and one speed. A tenth of a pixel and ten metres a second throughout; the arm narrows with the road read in the way the length figure shows, and a faster pass moves the pitch’s swelling to longer wavelengths, where the road’s power is greater.

Still open: whether the rows read where the body pitches about #

The arm here was measured from the body’s centre, and the rows read it well because a body was assumed to pitch about that point. A car pitches about a point its springs decide: stiffer front springs move it forwards, a heavy boot backwards. The rows read the camera’s distance from wherever that point is, and a second camera, or the same camera moved, would read a second distance from the same point.

The measurement that settles what that is worth puts the pitch centre somewhere between the axles, couples the bounce and the pitch as unequal springs do, and asks whether one pass reads the coupling as a sixth number, and whether a camera mounted at two known places along the body, read in turn, finds the pitch centre itself rather than only the camera’s distance from it.