A real road sets its own power

A wheel that rolls without slipping turns through its path length over its radius, and what a wheel counts on a rough road found that the path it counts is its centre’s. On a flat road that is the distance. On a corrugated one the centre rises and falls, the path is longer than the distance, and the wheel overcounts. That essay built its road from a single sine and found the overcount falling as the inverse square of the wheel’s radius once the wheel was large enough to bridge the troughs. A second wheel cannot correct the first turned that law into a recipe. Two wheels of different radii measure their own error, since the error is known to go as $1/(R + \rho)^2$: wheels of 0.6 and 1.2 m extrapolate to 0.018% on its road, knowing nothing else about it.

That essay ended on the one assumption everything rested on: the road repeated. A real road is not a sine. It is closer to a random surface with roughness at every wavelength, and a wheel’s centre path is that surface seen through the wheel’s size, short waves bridged and long ones followed. The question was whether the inverse square survives a spectrum, or whether the long waves no wheel bridges set a floor that no size of wheel and no extrapolation can get under.

Neither answer is quite right. The long waves do set a floor, and on the band road standards describe it is the whole of the error. Below that band the error falls with the wheel’s size, but as a power the road chooses, and on a real road it is two thirds, not two.

A road with a spectrum #

Roads are classified by the spectrum of their height. ISO 8608 describes a road’s height as a random signal whose power at spatial frequency n, in cycles per metre, falls as $n^{-2}$. The classes differ only in the level: class C, an average road, has $256 \times 10^{-6}\,\mathrm{m^3}$ at 0.1 cycles per metre. The standard describes the band from 0.011 to 2.83 cycles per metre, which is wavelengths from 90 m down to 0.35 m. Shorter than that, a road has texture, the grain of its surface, which the standard leaves to other measurements.

A road with that spectrum was built as a sum of sines, 60 or 80 of them at wavelengths spaced evenly on a logarithmic scale, each with a random phase and with the amplitude the spectrum gives its share of the band. Three different sets of phases give three roads with the same spectrum and different surfaces. Their height varies by about a centimetre, root mean square, mostly from the longest waves. Their slope is dominated by the shortest.

The road’s own excess length, over the distance, is half its mean square slope for gentle slopes, $\sum \pi^2 A^2/\lambda^2$ over the components. Over the ISO band it is $1.51 \times 10^{-4}$, a road 0.015% longer than the distance it covers; with texture down to 5 cm, $1.05 \times 10^{-3}$. The measured lengths of 300 m of each road agree with those sums to under 1%.

Measuring a wheel: the road dilated by a disc #

A wheel resting on a road at horizontal position X has its centre at the greatest height a circle of radius R can sit, which is the highest of $y(x) + \sqrt{R^2 - (x - X)^2}$ over the road points within R. Taken over every X, that is the road dilated by a disc, the operation image processing calls a morphological dilation, and it gives the centre path at once. It is the same operation as the room a machine sweeps, turned round: there a part was moved over a drive and its covered ground taken, here a disc is moved along a road and the highest ground it can sit on is taken. The road was sampled every 2 mm with texture, 5 mm without, and each centre height taken as that maximum over the samples. The overcount is the centre path’s length over 300 m, divided by 300 m, less one.

The earlier essays found each contact point by a fine search and a refinement, recomputing the road at every trial point. On a road of eighty sines that was eighty times slower, and a first attempt at the largest wheels also searched too coarsely to see 5 cm texture under a 10 m wheel, and got its answers wrong: the 4.8 m wheel came out with more excess than the 2.4 m one, because its search stepped over the texture’s crests and let the wheel sink between them. That is the failure a sweep that missed nothing warned of, a finite search reporting only what it looked at. Sampling the road once, every 2 mm, and dilating the samples is both faster and fine enough everywhere, since no wheel’s contact can fall between samples closer than a twenty-fifth of the shortest wave.

Over the standard band, the wheel does not matter #

Over the band the standard describes, the answer is flat. Wheels of 0.1, 0.3, 0.6 and 1.2 m all count exactly the road’s own excess, $1.51 \times 10^{-4}$, to 1%, on all three roads. Only a wheel of several metres begins to bridge anything: at 4.8 m the excess is 0.86 of the road’s, at 9.6 m 0.61.

The reason is the road’s tightest trough. A wheel that fits into every trough touches the road everywhere, and then its centre path is the road offset by R along its normal, which has the road’s length up to a term in the total turning that vanishes over a long road. The tightest concave radius of curvature anywhere on the three roads is 1.53, 1.66 and 1.35 m. Every wheel smaller than that counts the whole road, and every measuring wheel, bicycle wheel and car wheel is smaller.

So on a road described only by its standard band there is a floor, and it is the entire error: 0.015% for class C, and no practical wheel’s size changes it.

The floor is also exactly what the road is, which is worth separating from what the wheel does. A road longer than the distance it covers by 0.015% is a road with that much slope in it, and a wheel following it has travelled that much further along the surface. Whether that counts as error depends on the question. For a surveyor who wants horizontal distance it is error, and one no wheel can remove; for a vehicle’s wear, which goes with the surface travelled, it is simply correct. The earlier essays’ one-sine road had its troughs sized so that ordinary wheels bridged them, and that is what made the wheel’s size matter there. Two wheels of different sizes count the same, and the extrapolation of the earlier essay has nothing to extrapolate from.

With texture, the error falls — as a power the road sets #

The texture below the standard band is where wheel size matters, because its troughs are small. With components down to 5 cm, the road’s own excess rises to $1.05 \times 10^{-3}$, seven times the band’s, and a wheel of ordinary size bridges part of it.

A wheel of 0.1 m counts $1.04 \times 10^{-3}$, almost the whole road. At 0.3 m it counts $8.27 \times 10^{-4}$, at 0.6 m $5.81 \times 10^{-4}$, at 1.2 m $3.85 \times 10^{-4}$, then $2.48 \times 10^{-4}$, $1.59 \times 10^{-4}$ and $1.01 \times 10^{-4}$ at 2.4, 4.8 and 9.6 m. Each doubling of the radius divides the excess by about 1.55, where the one-sine road’s inverse square would divide it by four. Fitted over the four largest wheels the slope is −0.64. The three roads agree to within 2% at every radius.

Why two thirds #

A wheel follows a wave while the wave’s troughs are wider than the wheel. A sine of amplitude A and wavelength λ has troughs of radius $\lambda^2/(4\pi^2 A)$. On a road whose spectrum falls as $n^{-w}$, with components spaced on a logarithmic scale, the amplitude grows with wavelength as $\lambda^{(w-1)/2}$, so the trough radius grows as $\lambda^{(5-w)/2}$. That is faster than the wavelength itself. The shortest wave a wheel of radius R follows therefore has a length growing as $R^{2/(5-w)}$.

The road’s excess length, band by band, goes as $\lambda^{w-3}$. For any spectrum flatter than $n^{-3}$ it is dominated by the shortest waves present, so the wheel’s excess is dominated by the shortest waves it follows, and it falls as

$$e(R) \propto R^{-p}, \qquad p = \frac{2(3 - w)}{5 - w}.$$

For the ISO shape, w = 2, that is two thirds. The argument also says when the law breaks. A spectrum steeper than $n^{-3}$ puts the road’s excess in its longest waves rather than its shortest, and then no wheel of practical size touches the part that matters: every wheel would count nearly the whole road, as the standard band already does. At exactly $n^{-3}$ the exponent is nought, and the excess falls only as the logarithm of the wheel’s size. The ISO classification uses two throughout, which is the case measured here. On the textured road, the wheels of 0.3, 1.2 and 4.8 m follow every single component longer than 0.05, 0.11 and 0.28 m. The factor of 2.55 between the last two is $4^{2/3}$ to within its sampling.

A single sine is the case of one wavelength and no spectrum. Its wheel follows it, or bridges it, and once bridged the excess comes from the arcs the centre traces over the crests, which gives the inverse square. A road with a spectrum always has a next wave just longer than the last one bridged, and the excess is set by that wave rather than by the arcs.

The law, tested where it could fail #

A law derived for one spectrum is a guess about others, so roads were built with spectra falling as $n^{-2}$, $n^{-2.25}$, $n^{-2.5}$ and $n^{-2.75}$, and the exponent was fitted over wheels of 1.2 to 9.6 m on each. It came out 0.643, 0.520, 0.376 and 0.220, against the law’s 0.667, 0.545, 0.400 and 0.222. Every measurement falls a few hundredths below the law, the same amount on each road. The fit spans only a factor of eight in radius, and the rolled wheel still bridges some superposed troughs tighter than any single component’s, which the law counts per component. The trend is the law’s. The power is the road’s, set by its spectrum, and the wheel only chooses where on the spectrum it sits.

What the one-sine recipe does on a real road #

The earlier essay’s recipe takes two wheels, assumes $e = e_\infty + c/R^2$, and reads off $e_\infty$. On a road of one sine $e_\infty$ is the distance’s own error, nought, and the recipe works.

On the textured road it assumes the wrong power and removes little. From wheels of 0.6 and 1.2 m it concludes the error left is $3.20 \times 10^{-4}$, where the 1.2 m wheel’s own is $3.85 \times 10^{-4}$: it has removed 17% of the larger wheel’s error. From 0.3 and 0.6 m it removes 14%, from 2.4 and 4.8 m 19%. The true error of an infinitely large wheel is nought, since such a wheel bridges everything, and the recipe comes nowhere near it.

Knowing the power, a two-wheel correction could be rebuilt around it, but it would need the road’s spectral slope, which is a property of the road, not something two wheels can report. And over the standard band there is nothing to correct with wheel size at all. The honest statement for a road-measuring wheel is that it counts the road it follows. On an average road with texture that is 0.04 to 0.08% more than the distance for a wheel of 0.3 to 1.2 m, and no pairing of wheels on the same road removes much of it. The recipe fails in the way the essay on what a measurement cannot tell describes a model failing: its residual can be made to fit two points perfectly while the thing it was built to estimate is missing from the model altogether.

Two equal wheels on one frame #

The earlier essay found that two equal wheels on a frame count alike on a repeating road, whatever the wheelbase, because both centres run on the same curve. On a road that does not repeat they run on the same curve too — the road dilated by the same disc — shifted by the wheelbase. Over 300 m of textured road two wheels of 0.6 m radius 1.5 m apart count excesses of $5.809 \times 10^{-4}$ and $5.799 \times 10^{-4}$, differing by 0.2% of either. The difference is the two ends of the run, the stretch one wheel has covered and the other has not, and it shrinks with the distance. Averaging two equal wheels still buys nothing. Two wheels in different tracks, left and right, would run on different roads and could disagree by more; but they would be different realizations of one spectrum, and the three roads here agree with one another to within 2% at every radius, so the disagreement would be small beside the error they share.

What the random road leaves out #

A rigid wheel. A tyre deforms over texture and does not bridge it the way a rigid disc does. For a pneumatic wheel the texture below a few centimetres is absorbed in the tyre, and its count sits closer to the standard-band floor than a rigid wheel of the same size.

Gaussian roads. Random phases give a road whose heights are close to Gaussian. Real roads have potholes, joints and repairs, which are not in a spectrum and which a wheel bridges or falls into one at a time.

One class. Every road here is class C. Over the standard band, where every wheel follows everything, the excess scales exactly with the class’s level: a class A road’s is sixteen times smaller and a class E road’s sixteen times larger. With texture the scaling is weaker, since a rougher road also has tighter troughs and a given wheel bridges less of it. The exponent, set by the spectrum’s slope, is the same for every class.

A rigid mounting. A car’s wheel rides on a suspension, and the wheel is the coupler found that its centre follows a coupler path of its own as the suspension moves. What an odometer counts is the wheel’s turning, which the road decides through the contact whatever the suspension does, so the count here is unchanged; what changes is where the wheel’s load presses it into the texture.

Still open: a wheel whose size is its measurement #

If the overcount falls as a known power of the wheel’s radius, set by the road’s spectrum, then a wheel’s count is a measurement of the road’s spectrum as well as of its length. Three wheels of different sizes rolled together would give the power and the level at once.

The distinct argument would be that inversion, a calibration of the kind a calibration is a synthesis with more equations set up, with the road’s two numbers as the unknowns. The measurement would roll three rigid wheels of different radii over roads of several classes and spectral slopes, fit $e(R) = e_0 + cR^{-p}$ through the three counts, and ask how well p recovers the road’s w, how well the level recovers its class, and how far the fitted $e_0$ is from the standard-band floor. If it works to a class, a trolley with three wheels is a road profiler that measures nothing but turns. A wheel that cannot report its radius found that rolling ties an angle to a size. Here the size would be telling the angle something about the road.