An eye that pitches with the road keeps its rows

A scroll of a climbing road measures its grade drew a straight climb through two slits, one leaning ten degrees forward along the track and one leaning ten degrees back, and found that every point’s two drawings were parted down the roll by one number: $2fg\sin\varphi$, 7.47 px at a five per cent grade, whatever the point’s depth and whatever its height. The eye in that measurement stayed upright, the way a walker’s head does on a hill. It ended by naming the case it had not measured, which is the one a camera on a vehicle is actually in: an eye bolted to the car, pitching up with the road.

The guess recorded there was that pitching would add something — that the posts would lean, and that the row offset might pick up a term in each point’s height, so that a tall object’s two drawings would part by a different amount from a short one’s. The first half of the guess is right. The second is wrong in the most useful direction available: the row offset does not gain a term. It loses everything it had.

Riding the road, the rows agree again #

The model keeps all of the upright scroll’s numbers — the focal length of 430 px, the paper advancing 26 px for every metre walked, slits leaning $\varphi = 10°$ either way — and changes one thing. The camera looks across the track, so when the vehicle pitches up by the grade’s angle $\theta_g$, the camera turns about its own line of sight. Its slit, which was a vertical plane leaning forward along the track, is now a plane leaning forward along the road and tipped back from the vertical by the pitch. Each drawing is still found by searching for the moment whose slit contains the point.

The hero figure is the same four posts the upright scroll drew, 2.4 m tall at 9, 18, 36 and 72 m from a road climbing five per cent, each shown as its two drawings side by side with the disparity along the roll taken out.

The two drawings of each post now share their rows exactly — the search returns a difference of zero to the last bit, at every post. The disparity along the roll is exactly the level scroll’s, too, without the $\sqrt{1+g^2}$ stretch the upright eye picked up, because the paper advances along the road and the slit now lies along the road as well. As far as the rows and the disparities are concerned, the pitched scroll of a climbing road is indistinguishable from the level scroll of a level one.

The reason is short. An eye riding the road moves along a straight line — the road — and its camera’s up direction is square to that line. The row a point lands on is the point’s height above the eye in the camera’s own frame, over the reach along the slit, and the height in the camera’s frame does not change as the eye slides along a line it is perpendicular to. The two moments that draw a point are just two places on that line. The upright eye was climbing out of its own camera’s horizontal plane between the two moments, and that climb was the offset; the pitched eye never leaves it.

Put another way, the pitched eye draws the world as it looks from a vehicle that believes the road is level. Everything the level scroll computed carries over, applied to a world that has been tipped back by the grade’s angle about the direction across the track. Which is exactly why the posts lean: in the vehicle’s frame, a true vertical is not vertical.

The rows read the camera against the road #

A camera rarely rides the road perfectly. Suspension sags under a load, a mounting bracket is set at some angle, a car’s nose rises under hard acceleration. The row offset turns out to be a measure of exactly that — the angle between the camera and the road.

With the camera pitched by $\theta_c$ on a road climbing at $\theta_g$, the reach along either slit is still exactly $D/\cos\varphi$ for a point $D$ metres out, and the two drawings part by

$$\Delta v = 2f\sin\varphi\,\tan(\theta_g - \theta_c).$$

Nothing about the point is in it. An upright camera, $\theta_c = 0$, recovers the earlier essay’s $2fg\sin\varphi$ exactly, since $\tan\theta_g$ is the grade. A camera riding the road shows zero. A camera pitched past the road — nose up more steeply than the road climbs, as under braking on a descent — shows the offset reversed. The figure searches three points at depths from 10 to 207 metres and heights from two metres below the track to eight above it, and all three sit on the line to $3\times10^{-14}$ px.

This says what the upright scroll’s measurement really was. It was never a reading of the road’s grade as such; it was a reading of how far the road climbed relative to a camera that did not climb with it. The upright walker happens to hold a camera whose horizontal is true, so the relative reading and the absolute one coincide. A vehicle’s camera shares the road’s own idea of level, and its rows report the discrepancy between the two — which is useful, and different. A dash-mounted scroll whose rows drift from zero is a scroll whose camera’s pitch is drifting against the road: a load shifting, a bracket loosening, a vehicle pitching over a crest faster than its camera’s mounting can follow.

A post leans by its depth #

The grade has not left the drawings. The posts in the hero figure lean, and they lean alike in both drawings — the forward slit and the backward one tip a vertical by the same amount in the same direction — so the lean carries nothing the second drawing could remove.

The mechanism is the one the earlier essay’s continuation predicted. The pitched slit plane leans back, so the top of a post is caught a little later than its foot: the eye has to travel a further distance equal to the post’s height times the sine of the pitch before the tipped plane reaches the top. That extra travel is a shift along the roll, and a shift along the roll is drawn at 26 px a metre at every depth, while the post’s own height is drawn at $f\cos\varphi/D$ px a metre and shrinks with distance. The ratio of the two is the lean:

$$\frac{du}{dv} = -\frac{s\,D\sin\theta_c}{f\cos\varphi\,\cos\theta_g}$$

columns per row, the minus sign saying that the top runs ahead of the foot. It grows in proportion to the depth. At five per cent a post five metres out leans by less than a degree; one at 160 m leans by 26°. The searched leans match the closed form to $2\times10^{-14}$.

This is the familiar signature of a pushbroom in a new place. A scroll is a camera that moves established that the direction along the roll is a map, drawn at one scale for every depth, while the direction across it is a picture that shrinks with distance. Any feature that has extent along the roll therefore dominates, far off, any feature that does not, and a pitched slit gives a vertical post a small extent along the roll. Near, the post’s height wins; far, the shift wins. A map along, and a picture across measured that asymmetry on flat things; a leaning post is the same asymmetry standing up. Every row is a different camera found the same shape in a rolling shutter, where a vertical line leans by an amount set by the image speed at its depth. There the lean was a nuisance that had to be undone before anything could be read. Here it is the reading.

What the lean measures, taken alone, is the pitch times the depth. To turn it into a pitch, the depth has to come from somewhere, and the pair supplies one — the disparity along the roll is $2sD\tan\varphi$ exactly — so the lean divided by the disparity is $\sin\theta_c / (2f\sin\varphi\cos\theta_g)$ with no depth in it. The pair reads the pitch from the posts as cleanly as the upright scroll read the grade from the rows. That much was expected. What was not is that one drawing turns out to be enough.

One drawing holds the grade as a product #

A single drawing has no disparity, so it cannot supply a post’s depth. But a single drawing of a climbing road contains two kinds of line that depend on depth in opposite ways. A vertical leans more the farther away it is. A level line — an eave, a lake shore, the top course of a wall — slopes down the roll less the farther away it is. The earlier essay measured the second for the upright eye; for a pitched eye the slope is

$$\frac{dv}{du} = \frac{f\cos\varphi\,\sin\theta_g}{s\,D\,\cos\theta_c}.$$

The depth is in the numerator of one and the denominator of the other. Multiply them and it cancels:

$$-\frac{du}{dv}\Big|_{\text{vertical}}\times\frac{dv}{du}\Big|_{\text{level}} = \tan\theta_c\,\tan\theta_g.$$

For an eye riding the road the two angles are equal, and the product is the square of the grade. Take the ratio instead, and what cancels is the grade, leaving the square of $sD/(f\cos\varphi)$ — the depth.

The figure takes four houses from 12 to 96 metres beside the five per cent road and reads each from one drawing: the corner and the eave, nothing else. The nearest house’s corner leans two degrees and its eave slopes four; the farthest’s corner leans sixteen and its eave half a degree. Every house returns the grade to $3\times10^{-16}$ and its own distance to $9\times10^{-13}$ m. Each house is scaled to fit its panel, which changes no angle.

This is new. The upright eye’s single drawing could not read the grade, because its posts were exactly vertical and its level lines’ slope needed a depth to interpret. A straight line in a scroll is a hyperbola found that a single scroll’s straightness is a statement about depth; on a pitched climb its angles are too, and a pair of angles at one depth settles both. The price is that the two lines must be at the same depth, which is why the figure uses a house: a corner and its own eave are the commonest pair of a vertical and a level line guaranteed to share a distance.

One caution belongs here. The level line must be genuinely level. A fence rail or a kerb built along the road climbs with it, and a line that climbs with the road is drawn exactly flat by a pitched eye — it is level in the vehicle’s frame — so its slope is zero and the product reads a grade of nothing. An eave is level because buildings are, whatever the road does. A reader of a real pitched scroll has to know which of the two a line is, and the geometry offers no way to tell from the line alone.

The product and the difference return both angles #

For a camera that does not ride the road exactly, the single drawing’s product is $\tan\theta_c\tan\theta_g$ and the pair’s rows are $\tan(\theta_g - \theta_c)$. Two equations, two unknowns. The product and the difference together return both angles.

The figure reads six vehicles this way, from a house thirty metres out and the rows of one point beside it, and each returns both its road’s grade and its camera’s pitch to the precision of the arithmetic. The six are chosen to show which mark carries what. On the upright camera the corner does not lean, the product is zero, and the rows carry the whole grade. On the camera that rides the road the rows are zero and the product carries it. On a level road with the camera nose-up, the corners lean but the eaves are level, so the product is zero again, and the rows now carry the whole pitch, reversed. Every mark vanishes somewhere, but never all at once.

Solving the two equations leaves a sign to choose, since the product of two tangents cannot tell a climb seen by a nose-up camera from a descent seen by a nose-down one. The eave settles it: its slope has the sign of the road’s grade, so a level line falling ahead is a climb and one rising ahead is a descent. The riding four-per-cent descent in the figure comes back as a descent for exactly that reason.

The same structure appeared in a scroll can be asked its own radius, where a bend’s radius came back only from two marks together — the disparity and the scale beside it — and neither alone could give it. A pitched climb is a second case of a scroll whose single measurement is a product and whose pair supplies the missing factor. The upright scroll was the special case in which one of the two factors was known to be zero.

What a pixel costs one drawing #

A reading from one drawing is only worth having if it is precise enough to use. Its precision has an unusual shape, because the two lines it rests on fail in different places.

The corner is the robust half. Its top runs ahead of its foot by $sH\sin\theta_c$ columns — 7.79 px for a six-metre corner at five per cent — and that number does not depend on how far away the house is: the extra travel is a length along the roll, and the roll is drawn at one scale everywhere. A pixel of error in reading that shift is the same fraction of it near and far. The eave is the fragile half. Its drop over twelve metres of run is 21 rows at twelve metres and barely more than one row at 192, and a pixel’s error in a drop of one row is an error of a hundred per cent.

So one drawing’s reading is good nearby and poor far off. At twelve metres a house read to a pixel gives the grade to ±0.49 per cent, better than the ±0.67 an upright pair gives from one point read to the same pixel. At 48 m it gives ±0.80, and at 192 m, ±2.71. The upright pair’s precision does not depend on depth at all, since every point reports the same row offset. The two instruments are different in kind: the pair is uniform across the picture, and a single drawing of a pitched climb is a near-field instrument whose reach is set by how long the level lines in it are.

The remedy is the obvious one. A longer eave, or a lake shore running the length of the picture, gives a larger drop to read; a taller corner — a church tower rather than a house — gives a larger shift. The figure’s numbers are for one ordinary house read to one pixel, which is the least favourable case a real picture offers.

Why the upright eye’s reading was relative all along #

The measurement in the earlier essay looked like a direct reading of the road: the rows parted by the height climbed over the reach. The pitched eye shows what it was a reading of. Rows are heights in the camera’s frame. They measure how far the eye rose relative to its own horizontal, and an upright walker’s horizontal happens to be gravity’s.

The centre a scroll does not have found that a scroll’s rays miss any common centre by the spread of the eye’s track, and a scroll through two slits ranges in a straight line turned the along-track part of that spread into depth. On a climb the track spreads upward too, and whether that upward spread shows in the rows depends only on whether the camera shares it. A camera that climbs with its own track sees no climb, in the same way a passenger in a train sees no motion in the carriage. The climb is still in the picture, but it has moved into the geometry of the world as the vehicle sees it — tipped by the grade — and that tipped world shows itself in the angles of anything the vehicle’s frame does not share: true verticals and true levels.

That is also why a single drawing can read the grade only for a pitched eye. An upright eye’s frame agrees with gravity, so verticals are vertical and nothing in a single drawing records the grade except level lines, whose slopes need a depth. A pitched eye’s frame disagrees with gravity by exactly the grade, and every vertical in the picture is a small plumb line showing by how much.

What the pitch leaves alone and what it does not #

Depth. The disparity along the roll is exactly the level scroll’s, $2sD\tan\varphi$, for every pitch. Depth is a reciprocal found a pinhole pair’s depth error growing with the square of the distance; this one’s stays flat, as the level scroll’s did. A riding eye gives the depth of every point without the $\sqrt{1+g^2}$ correction the upright eye needed, because its paper and its slit both follow the road.

Heights above the road. The rows a pitched eye draws are heights above the road’s own line, not above a level one. A row on a pitched scroll says how high a point stands off the road surface’s extension at that distance, which on a climb is less than its height above sea level by the grade times its distance along. A reader wanting true heights needs the grade first, and the corner and eave are how to get it.

Bends. Everything here is on a straight road. A pitched camera on a climbing bend is a helix drawn by a tipped slit, and the helix’s saturation — the earlier essay found its row offset falling with depth as the two moments close up — will change how the lean depends on depth. That case is not measured.

Still open: a vehicle pitches with its wheelbase, not with the road #

A vehicle does not pitch with the road at the point where its camera is. It pitches with the chord of the road between its front and rear wheels, so on a crest or a dip its pitch is the grade averaged over one wheelbase, and it lags the grade under the camera by half a wheelbase on either side of a change. A scroll of a climbing road measures its grade found that each point’s rows report the grade as the chord between the eye’s two moments, $2D\tan\varphi$ apart. A pitching vehicle adds a second chord, one wheelbase long, and the rows now report the difference between two chords of different lengths.

On a steady grade the two chords agree and the rows are zero. Over a vertical curve — the parabola a road engineer lays over a crest — a chord’s slope is the grade at its midpoint, so both chords report the grade at the same place and the rows stay zero again. The difference appears only where the grade changes abruptly, and there it is a statement about the road’s curvature measured over two lengths at once. The measurement that settles what that is worth runs a pitched scroll over a step in grade with a stated wheelbase, and asks whether the rows’ excursion — how far they depart from zero and over what stretch of roll — returns the wheelbase and the size of the step, and whether a point near enough to have a short chord of its own reads the step more sharply than the vehicle’s pitch can follow it.