A scroll of a climbing road measures its grade

A scroll through two slits ranges in a straight line drew a handscroll twice from one level track — once through a slit leaning ten degrees forward along the track and once through a slit leaning ten degrees back — and found every point drawn twice on the same row, separated along the roll by a distance proportional to its depth. Everything built on that pair since has used the shared row. A scroll round a bend loses its straight-line depth bent the track and kept it; a scroll can be asked its own radius recovered the bend from two marks and, at its end, named the one assumption none of them had tested: the track was level.

The shared row is a consequence of that assumption. A point is drawn by the forward slit from one position of the eye and by the backward slit from another, and its row in each drawing is the eye’s height above the point divided by the distance to it. On a level track the two eye positions are at one height and the two distances are equal, so the rows agree. On a road that climbs, the second position is higher than the first.

The guess the earlier essay recorded was that this might add noise, or might add a measurement. The measurement below says which, and it says so with an unusually clean number.

Every post drops by the same amount #

The model keeps everything the level scroll had and changes one thing. The eye stays upright, so its slit is still a vertical plane leaning φ = 10° toward the direction of travel, and the row a point lands on is still the divide inside that plane: the focal length, 430 px, times the eye’s height above the point, over the reach along the slit. The paper still advances with the journey, now by the distance walked, which on a slope is a little more than the distance travelled over the ground. Only the eye’s height changes: it rises by the grade $g$ for every metre travelled.

The hero figure takes four posts at 9, 18, 36 and 72 m from a road climbing five per cent, draws each through both slits, and sets each post’s two drawings side by side with its disparity taken out. The disparities are what the level scroll had, times a factor the next section accounts for: 82.6, 165.2, 330.5 and 661.0 px along the roll. What remains between each pair is a drop down the roll, and it is 7.47 px at every post — the post nine metres out and the one seventy-two metres out alike.

The number is a closed form with nothing about the point in it. Between the two moments that draw a point $D$ metres out, the eye travels $2D\tan\varphi$ along the road and so climbs $2gD\tan\varphi$. The reach along either slit is $D/\cos\varphi$. The row difference is the focal length times the climb over the reach:

$$\Delta v = \frac{f \cdot 2gD\tan\varphi}{D/\cos\varphi} = 2fg\sin\varphi.$$

The depth cancels because the climb and the reach both grow in proportion to it, and the point’s height never enters, because it is subtracted from both eye heights alike. For a five per cent climb that is 7.467 px; for two per cent, 2.987; for ten, 14.934. Every figure below finds its drawings by a search for the eye position whose slit contains the point, never by this formula, so each agreement is a check rather than a restatement.

A bend does not do this, and a climbing bend does something else #

The question the level scroll left was whether a climbing straight track and a level bend might leave the same marks, so that no reader could tell them apart. The row offset settles it before anything else is measured.

On straight climbs of two, five and ten per cent, points from five to 320 metres out and from ground level to twelve metres up all show the same offset, to $3 \times 10^{-13}$ px. On a level bend of 200 m the offset is exactly zero: the two eye positions sit symmetrically about the point’s bearing, at the same height, at the same reach. A bend moves the columns and never the rows, and a climb moves the rows. The two are not confounded.

A road that bends and climbs together — a helix, in plan a circle of 200 m and in section a five per cent grade — does something neither does alone. Its offset is 7.28 px for a point 5 m out and falls to 2.88 px at 320 m. The reason is the bend’s own saturation, which a scroll round a bend found in the columns: outside a bend the two slits’ moments close up toward a fixed angle however deep the point, so the climb between them stops growing, while the reach keeps growing with depth. The offset is the same climb-over-reach, now with a numerator that saturates. On the helix the climb’s mark carries depth information, where on the straight road it carried none.

One drawing alone shows the grade as a lean #

The pair is not the only place the grade shows. A single drawing of a climbing road already carries it, in a form the eye picks up at once.

As the eye climbs, everything level beside the road falls down the page, and it falls faster the nearer it is. A level line $D$ metres out slopes down the roll by $fg\cos\varphi / (D s\sqrt{1+g^2})$ rows per column, where $s$ is the paper’s 26 px per metre. On a five per cent climb that is 9.24° for a kerb 5 m away, 4.65° at 10 m, 1.16° at 40 m and 0.23° at 200 m: each line leans twice as much as one twice as far. A line that climbs with the road — a wall built along it, a handrail — is drawn exactly level, because it rises as fast as the eye does.

This is the same fact seen once rather than twice. In a single drawing, the grade appears as a lean that scales with nearness, and reading it needs the depth, which one drawing does not have. In the pair, the depth is supplied by the disparity and cancels out of the row offset, which is why the pair’s measurement needs nothing about the point. A reader of one drawing sees a road climbing through a landscape that tilts; a reader of two reads the grade in pixels.

The lean has one consequence worth stating for anyone reading a single scroll. A level shoreline in the distance and a level kerb close by lean by different amounts, and the difference is not an error in the painting — a straight line in a scroll is a hyperbola already found that straightness in a scroll is a statement about depth, and on a climb the slope of a level line is too.

The grade is read over a stretch that widens with depth #

A real road does not climb at one grade. It levels out, steepens, crests. The row offset follows, but not point by point.

Each point’s two drawings are made from two positions $2D\tan\varphi$ apart, so what its row offset reports is the height the eye climbed between them over that distance: the slope of the chord of the road’s profile across that stretch. For a point 10 m out the stretch is 3.5 m and an abrupt change of grade reads nearly as a step. For a point 160 m out it is 56.4 m and the same change is spread into a ramp that wide. The reading matches the chord to $6 \times 10^{-16}$.

Two things follow. A scroll can map a road’s grade along its length, but the resolution of the map is set by how far away the points used are: near objects give sharp readings, the far hills a blurred one. And a road whose grade changes steadily — the parabolic vertical curve a road engineer lays over a crest — is read exactly, at every depth, because a chord of a parabola has the slope of the parabola at its midpoint, and the midpoint of the two moments is directly across from the point. The blur appears only where the grade changes abruptly.

A climbing bend gives back all three #

The level bend was recovered from two marks per point: the separation of its two drawings and the along-roll scale beside it. The helix has a third, the row offset, and a third unknown, the grade.

The three marks do not separate in one step, but they nearly do. The bend’s closed form needs the paper’s scale per metre of ground, and the paper follows the slope, so that scale is stretched by $\sqrt{1+g^2}$ — the only way the climb reaches the columns at all. The grade needs the bend’s angle from the columns. Read together, alternately, they settle in seven steps, and every point from 5 m to 120 m returns a radius of 200 m to $2 \times 10^{-11}$ m, its own depth to $4 \times 10^{-13}$ m, and a grade of five per cent to $2 \times 10^{-16}$.

The more instructive row of the figure is the reading that ignores the climb. The stretch is 0.125 per cent of the paper’s scale at a five per cent grade — nothing, to the eye — and a reader who treats the helix as a level bend recovers every depth within 0.002 per cent. The radius is another matter: 200.67 m from the point 120 m out and 210.78 m from the one 5 m out. A scroll can be asked its own radius found that the radius multiplies any error in the along-roll scale by about $R/D$, which is 40 for a point 5 m from a 200 m bend; forty times an eighth of a per cent is five per cent. The climb costs the bend nothing it can see and five per cent of its radius, and the third mark is what pays that back.

What a pixel of the offset is worth #

A reading that needs nothing about the point is only useful if it is precise, and the precision is a closed form too.

A pixel of error in one offset is a grade error of $1/(2f\sin\varphi)$: 0.67 per cent, checked against four thousand seeded draws at 0.677. That is enough to tell a two per cent climb from a five per cent one from a single point, and on a straight road every point measures the same number, so averaging helps exactly as independent readings should: sixteen points read to half a pixel give the grade to ±0.084 per cent. The depth the same points give is held to 0.109 m a pixel, as on the level track, stretched by $\sqrt{1+g^2}$ — an eighth of a per cent at a five per cent grade, and correctable once the grade is known.

The contrast with the level scroll’s other readings is the useful part. The bend’s radius was fragile, amplified by $R/D$; the depth was robust. The grade is robust in the way the depth is, because it rests on a difference of two rows that nothing about the point disturbs, and it adds a fact about the journey — how steep it was — that a level scroll never contained.

The roll measures the road, not the map #

The stretch that cost the naive bend its radius says something about the level scroll’s most useful property as well. A map along, and a picture across found that the direction along the roll is an exact scale drawing: the midpoint of a segment lying along the scroll images to the midpoint of its image, and distances along the roll are distances along the ground at 26 px a metre. On a climb that stays true with one change of meaning. The paper advances with the walk, so distances along the roll are distances along the road — 26 px for every metre walked up the slope — and the plan distance is shorter by a factor of $\sqrt{1+g^2}$.

At a five per cent grade the difference is an eighth of a per cent, and no one reading a painted journey would notice it. What matters is which quantity the scroll is a scale drawing of. A handscroll of a mountain path is a map of the path as walked, not of the ground it crosses, and two points a known distance apart on the map will be drawn slightly too far apart on the roll wherever the path between them climbs. The row offset is what lets a reader convert one into the other, since it gives the grade at every stretch of the roll.

The contrast with a pinhole pair is worth stating. Depth is a reciprocal found that a pinhole pair’s depth error grows with the square of the depth; the two-slit scroll’s depth error does not grow at all, and neither does its error in the grade, since every point, near or far, reports the same row offset with the same precision. On a straight climb the scroll reads both the ground’s depth and the road’s slope uniformly across the whole picture.

Why a hill adds a mark rather than noise #

The level scroll’s rows were never uninformative. They were redundant: two drawings of a point that agree in their row carry one row between them, and the second was a check on the first. A climb spends that redundancy. The row the second drawing lands on now differs from the first by exactly the height climbed over the reach, and since the reach is fixed by the depth and the depth by the columns, the difference is a measurement of the climb with nothing left over.

That is why a straight climb gives the same offset at every point: every point is measuring the same thing, the height climbed per metre travelled, and the two slits have chosen a baseline and a reach whose ratio does not depend on the point. The centre a scroll does not have showed that a scroll’s rays miss any common centre by the spread of the eye’s track; on a climb that track spreads upward as well as along, and the two slits turn the upward spread into a number on the page.

What this leaves out #

An eye that tilts with the road. The eye here stays upright, as a painter walking uphill does. A camera fixed to a vehicle pitches with the grade, and its slit is then no longer vertical; which eye position sees a point starts to depend on the point’s height, and the two drawings’ columns acquire a term in it. That case is not measured here, and it is the continuation below.

Where the paper’s scale comes from. The paper here advances by the distance walked. A scroll unrolled by the distance over the ground has no stretch, and its reading of the bend is unaffected by the climb; the grade is read the same way under either convention.

Grades that change within the stretch a point sees. Every reading is a chord. A profile with structure shorter than $2D\tan\varphi$ — a hump, a step in the road — is averaged away for points farther out than that structure, and the figure above is the extent of it.

Still open: an eye that tilts with the road #

A camera on a vehicle climbing a grade pitches up by the grade’s angle, and its scan slit — the line of pixels it records at each moment — pitches with it. The slit plane then leans back from the vertical by that angle, so a tall object is no longer recorded all at one moment: its top is recorded a little before or after its foot, and every vertical post in the scroll leans.

The measurement that settles what that costs pitches the eye with the road, draws the same posts through both slits, and asks three things: how far a vertical post leans in each drawing as a function of its depth and the grade, whether the row offset stays the same for every point once the pitch is included or acquires a term in the point’s height, and whether the lean of the posts is itself a second reading of the grade — one that a single drawing, without its pair, could give.