A vehicle's pitch lags the road by its wheelbase

An eye that pitches with the road keeps its rows fixed a scroll camera to a vehicle climbing a straight grade and found that the row offset a scroll of a climbing road measures its grade had read — each point’s two drawings parted down the roll by the height climbed between the two moments — vanished exactly. An eye that pitches with the road it climbs carries its whole frame up the grade, and a point’s two drawings share their rows again. The grade had moved into the posts, which leaned.

That essay ended on the approximation hidden in “pitches with the road”. A vehicle does not pitch with the road at the point its camera is over. It pitches with the chord between its front and rear wheels, so on a change of grade it lags: the rear wheels are still on the old grade while the front wheels are on the new one, and for a wheelbase’s worth of travel the vehicle is part-way turned. The rows read the difference between the grade the eye’s two moments straddle and the grade the vehicle is pitched to, and wherever those two chords of the road differ, the rows should come back.

They do, and they come back as a signature whose width says how long the vehicle is.

Two chords of one road #

The scroll here is the same instrument as before: a camera travelling along a straight road, drawing each point through two slits leaning ten degrees forward and ten back, so that a point is drawn twice, at two moments some distance apart. For a post twelve metres from the track that distance — the eye’s own chord — is $2D\tan\varphi$, 4.23 metres. The road is level up to a point and climbs six per cent after it. The camera rides the road at eye height and is pitched to the chord between wheels a stated distance apart, centred under it.

On the level road and on the steady six per cent the rows are zero, exactly as the earlier essay found: on a straight grade every chord is the grade, so the vehicle’s pitch and the eye’s climb agree. Near the step they part. A camera pitched with the road at the very point it stands on — a vehicle with no wheelbase — parts a post’s rows by up to 4.26 as the post’s two moments straddle the step, over the four metres of road in which they can. A camera on a 2.7-metre wheelbase parts them by up to 2.18, over 6.5 metres; on a 4.5-metre wheelbase, 2.12 over 8.5. Drag the middle vehicle’s wheelbase and its curve moves between the other two: at 1.5 metres the rows part by 2.75 over 5.5 metres of road, at 3.9 metres by 2.13 over 8.0. The height settles quickly once the wheelbase passes the eye’s own chord; the width keeps growing with it, which is the fact the next section reads.

The excursion runs one way before the step and the other way after it: the rows change sign as the post passes the change of grade, because a post just before it has its two moments split across the step one way round and a post just after it the other. Far from the step both chords lie on one grade and there is nothing to part them.

The width says the wheelbase #

The height of the excursion depends on how big the step is and how the two chords overlap. Its width is simpler, because it is set by lengths along the road alone.

A post’s rows depart only while one of the two chords straddles the step: the eye’s, between its two moments, or the vehicle’s, between its wheels. For a camera pitched at the point there is only the eye’s chord, and the excursion is exactly as wide as it is, from a metre for a post three metres out to fourteen for one forty metres out. On a vehicle the vehicle’s chord adds its own length, and the excursion widens by about the wheelbase at every depth: 2.5 to 3.0 metres against a true 2.7, read at the quarter-metre spacing the posts here are placed at.

That is a measurement of the vehicle from its own pictures. A scroll through two slits ranges in a straight line found that the two slits’ disparity gives every point’s depth; the rows now give, from the same pair of drawings, how long the machine carrying the camera is. A post near the track has a short chord of its own, so most of its excursion’s width is the vehicle’s; a post far out has a long one, and the wheelbase is a small part of what it shows.

A vertical curve does not silence the rows #

Roads are not built with steps in their grade. A road engineer eases a change of grade with a vertical curve, a parabola laid between the two straight grades, and the earlier essay argued that on a parabola the rows should vanish: a chord’s slope on a parabola is the grade at its midpoint, so two chords centred on the same place report the same grade, whatever their lengths.

The argument is right inside the curve and wrong at its ends. While both chords lie entirely on the parabola, they report the same grade and the rows are zero. Where either chord reaches past the curve’s end onto a straight grade, it is no longer a chord of a parabola, and the two chords disagree by an amount set by how fast the grade was changing — the step divided by the curve’s length. So the rows do not vanish; they fall. An abrupt step leaves 2.18 of them; a two-metre curve, 1.85; five metres, 0.94; ten metres, 0.47; forty metres, 0.12, falling as about 4.7 divided by the curve’s length once it is longer than the chords.

A real road’s vertical curves run from tens of metres on a residential street to hundreds on a highway, so on real roads the excursion is a few tenths of a row or less, confined to the curve’s two ends. It is readable where the curve is short — a driveway’s lip, a kerb ramp, a railway crossing’s hump — and those are exactly the changes of grade a vehicle’s suspension notices.

When the wheelbase stops mattering #

The height of the excursion depends on the wheelbase against the eye’s own chord, and the dependence is not what the step figure suggests at first sight.

With no wheelbase the camera turns by the whole step the instant it crosses it, between one of a post’s moments and the next, and the rows depart by 4.26. A wheelbase makes the turn gradual: the vehicle turns over a wheelbase’s worth of travel, and a post whose two moments fall inside that stretch sees only part of the turn between them. The excursion halves as the wheelbase grows towards the eye’s own chord, 4.23 metres for these posts, and past that it hardly changes — 2.18 at 2.7 metres, 2.12 at 4.5, 2.14 at six. Once the wheelbase is longer than the eye’s chord, the vehicle is always part-turned while a post is being drawn, and what sets the rows is the eye’s own chord against the step.

That gives the practical reading of the earlier essay’s claim. An eye riding the road keeps its rows on any steady grade, whatever the vehicle. On a change of grade, a short vehicle behaves like an eye that pitches at the point and parts the rows sharply; a long vehicle parts them about half as much, over a stretch as long as itself. Either way the rows’ excursion is a record of the change of grade, drawn in the scroll at the place it happened.

One line of posts reads the vehicle #

A record is worth having only if it can be read back, and the excursion has two things in it: how big the step was, in its height, and how long the vehicle is, in its shape.

Reading a line of posts every quarter-metre along twenty metres of road, twelve metres out, with each row read to a tenth of a pixel, the wheelbase comes back as 2.70 ± 0.07 metres and the step as 5.99 ± 0.07 per cent. At four tenths of a pixel, 2.70 ± 0.19 metres and 6.01 ± 0.32 per cent. The fit is simple: for each candidate wheelbase the excursion’s shape is known, and its height scales with the step, so the step comes from a least-squares amplitude and the wheelbase from whichever shape leaves the smallest residual.

The two are separable because the posts sweep the step past both chords. A post is drawn at two moments, and at each moment the vehicle’s wheels are somewhere relative to the step; moving the post along the road slides both chords across the step at the same rate, and the excursion’s shape records where each chord begins and ends. Every row is a different camera is the principle underneath: a scroll is a sequence of cameras, and a vehicle’s pitch at each moment is one of those cameras’ orientations, recorded whether or not anyone meant to record it.

Why the two chords are the right description #

It is worth being exact about where the two chords come from, because the description is simple enough to be mistaken for a metaphor. A scroll is a camera that moves: each column of the roll is drawn by the camera at one moment, from one place on the road, pointing one way. A point beside the road is drawn twice, once by each slit, and the two drawings come from two moments separated by the distance along the road that the two slits’ leans put between them — for a point at depth $D$ and slits leaning $\pm\varphi$, a distance of $2D an\varphi$.

The row a point lands on depends on two things at each moment: how high the camera is relative to the point, and how the camera is pitched. Between the two moments the camera has climbed by the road’s rise over that distance — the chord of the road between the two moments — and it has turned by whatever the vehicle turned. The rows part by the climb minus what the turn takes back. For an eye pitched to the grade it stands on, on a straight grade, the turn takes back the climb exactly, as the earlier essay found. For a vehicle, the turn at each moment is the chord between its wheels at that moment, so what the rows record is the difference between a chord of length $2D an\varphi$ and the change, between the two moments, of a chord of length $L$. Where the road is straight both vanish together; on a parabola the chord slopes vary linearly and cancel; anywhere else they leave a trace.

That also says why the scroll has no single centre to fall back on. The centre a scroll does not have measured how far a scroll’s rays miss a common point, and found the miss set by the eye’s own track. On a changing grade the track bends in elevation, and the camera’s pitch follows the bend late; the rows are the part of that lag a pair of drawings can see.

What the posts have to be #

The reading back used posts every quarter-metre, twelve metres from the track. Real roadsides are not that obliging, and the two numbers the posts give behave differently as the posts thin out.

The step’s size comes from the excursion’s height, and any post whose two moments straddle the change of grade contributes to it; a handful of posts near the step fixes it nearly as well as a dense line. The wheelbase comes from the excursion’s shape and width, and those are read only as finely as the posts sample them: the width figure’s differences of 2.5 to 3.0 metres against a true 2.7 are the quarter-metre spacing showing through. A fence, a line of railings, the joints of a crash barrier or a row of lamp posts every few metres gives a spacing of that order; a hedge with no marks gives nothing. Posts at several depths help twice, since each depth has its own chord and so its own width, and the difference between their widths is the same wheelbase read again.

And the posts have to be where the step is. A change of grade leaves its signature over a stretch of road a few metres longer than the eye’s chord plus the wheelbase, and a scroll whose roadside is bare along that stretch records the change nowhere — the rows, like every reading in this sequence, are evidence only where there is something to draw. A scroll is not a panorama made the point that a scroll’s geometry is set by the track and not by any centre; the same track’s grade is read only where the roadside gives the scroll something to draw twice.

What the rows have been measuring #

The sequence that led here has been reading one quantity under different names. A scroll of a climbing road measures its grade found the rows reading the grade, as the chord between the eye’s two moments; the pitched eye found them reading the difference between the grade and the eye’s pitch; this essay finds that the pitch itself is a chord, one wheelbase long, so the rows read the difference between two chords of the road, of lengths $2D\tan\varphi$ and $L$. On a straight road the two agree; on a parabola they agree inside it; everywhere else their difference is the road’s change of grade, averaged over two different lengths.

A scroll can be asked its own radius found a bend in plan recoverable from the scroll’s columns; this is the same kind of result in elevation. A scroll camera on a vehicle records the road’s curvature in the vertical plane through the rows, and its own length as the scale over which it smooths that curvature. A vehicle is, in that sense, a low-pass filter on the road’s profile, and the rows are its output minus the eye’s.

What was assumed #

The vehicle is rigid and its suspension does not move. A real vehicle pitches on its springs as it brakes, accelerates and meets a change of grade, adding a pitch of its own that the chord does not predict. On a step, the suspension’s response is a damped oscillation after the step, which would show as a ringing in the rows past the excursion; the measurement here has none.

The camera is at the middle of the wheelbase. A camera mounted ahead of the middle rises and falls with the pitch as well as turning with it, adding a height change the eye’s own chord already reads, and shifts the vehicle’s chord relative to the eye’s. The excursion’s width still gains the wheelbase; its shape becomes lopsided.

The posts stand at a known distance from the track. The eye’s chord is set by each post’s depth, which the scroll’s columns give; a line of posts at mixed depths has excursions of different widths, and the fit must use each post’s own chord.

Still open: whether the suspension’s ringing can be read from the rows #

The rigid vehicle here turns smoothly over its wheelbase. A real one bounces: it meets a step with its front wheels, its springs compress, and it pitches past the chord and back, a damped oscillation of a second or so. The camera’s pitch then departs from the chord by an amount that rings down after each change of grade, and the rows, which read the camera’s pitch against the eye’s chord, should ring with it.

The measurement that settles it gives the vehicle a pitch that follows its chord through a damped spring of stated frequency and damping, drives it over a step at a stated speed, and asks whether the rows past the excursion show the ringing above the reading error — and if they do, whether its period and decay can be read back, which would make a scroll camera a record not only of the road’s profile and the vehicle’s length but of how the vehicle’s suspension was set.