Which way did the bicycle go

A bicycle has been ridden across soft ground and gone. What is left is two tracks, crossing each other where the rider steered hard and running close together where they did not. Nothing else — no rider, no machine, no witness.

Three questions can be asked of those marks, and all three have answers that follow from one line of the previous essay: which track was made by the front wheel, which way the bicycle was travelling, and how long the bicycle was.

The relation the tracks are in #

The rear wheel of a bicycle is a towed axle. Its rod is the wheelbase, its hitch is the front wheel’s contact point, and it obeys the condition that decides everything here:

the rear wheel points at the front wheel, always.

That is not an approximation or a tendency. It is the geometry of a rigid frame with the rear wheel’s axle perpendicular to the frame, and it holds at every instant of every ride regardless of speed, lean, rider or surface.

Its consequence for the marks is immediate. The rear track’s tangent, at any point, points along the frame; and the front wheel is exactly one wheelbase along that tangent, in the direction of travel. So the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Every time.

The reverse construction has no reason to work. The front wheel does not point at the rear wheel — it points wherever it is steered — so the front track’s tangent extended backwards lands wherever the steering happened to be pointing, which is not where the rear wheel is except by coincidence.

What the measurement says #

The test is run on a computed pair of tracks: a front path with two superposed sinusoids, and a rear path produced by the heading equation with a wheelbase of 1.05 m. At every sample of the candidate rear track, the tangent is taken by a central difference, extended by 1.05 m, and the distance from the landing point to the nearest point of the candidate front track is recorded.

The right way round: mean miss 0.13 mm.

The wrong way round: mean miss 447 mm.

A ratio of 3,554, on tracks whose own separation is a metre. This is not a close call and it is not a judgement; it is a number three orders of magnitude away from the other number.

The residual on the correct assignment is not zero, and what it is worth knowing. It is the central difference’s own error in taking the tangent — the tracks are sampled at finite spacing, and a chord is not a tangent — so it shrinks as the sampling is refined, and it would shrink to the rounding floor for a smooth track sampled finely enough. On real tracks in real mud it would be the width of the tyre and the raggedness of the edge, which is a much larger number and still nowhere near 447 mm.

What the tracks are asked and what they answer #

Three answers come out of one construction, and it is worth separating them because they have different strengths.

Which track is the front one. The strongest answer. The ratio above is enormous and it does not depend on knowing anything in advance.

Which way the bicycle went. The tangent has two directions and only one of them lands on the front track: extend the rear tangent the other way and it misses by as much as the wrong assignment does. So the direction of travel comes out of the same test, and it is the answer to the question the essay is named after.

How long the bicycle was. The wheelbase does not have to be known. Sweep it and take the value that minimises the residual, and what comes back is 1.0503 m against the 1.0500 the tracks were made with — recovered to three tenths of a millimetre from marks that were never told it. That is a small identification problem of exactly the kind an arm’s calibration is, with one parameter instead of thirty and the same structure: a geometric property is fitted to measurements of where something went.

The three-tenths of a millimetre took one correction, and it is the kind that is easy to leave in. The first version of the fit swept 240 values between 0.2 m and 3.0 m and returned the best of them, which is a grid of 11.7 mm — so it reported 1.0517, and the 1.7 mm of error was the sweep’s own resolution being quoted as a measurement. Adding a golden-section refinement inside the bracketing interval costs nine more evaluations and takes the answer to 1.0503, and the residual at the refined value is a third of the residual at the grid point. A quantity recovered from data has to be recovered to better than the machinery’s own step, or the number being published is a fact about the machinery.

The claim this replaces #

The question has a literary history. In The Adventure of the Priory School, Holmes reads a direction of travel from a bicycle’s tracks and explains that the rear wheel’s track is deeper because it carries more weight, so the deeper track overlying the shallower one shows which way the machine went.

The first half is true — a bicycle does carry more weight on its back wheel — and the second half is a different claim that does not follow from it. Depth says which track is which; it says nothing about direction, since the rear track overlies the front track at a crossing whenever the rear wheel arrived at that spot later, and which wheel arrives later at a given crossing depends on the shape of the path rather than on the direction of travel.

What is worth noticing is that the correct answer is available in the same marks and is stronger. The tangent test is geometric rather than physical: it needs no assumption about weight distribution, no assumption about which tyre is wider, and no assumption about the ground. It works on tracks in snow, in sand and in paint. And it returns a number rather than an impression, which means it can be wrong in a way that shows.

That is the pattern this site’s wrong field keeps finding: the confident reading is not usually a fabrication, it is a true statement pressed into answering a question it does not answer, with the question that it does answer left unasked.

What the tracks cannot say #

An honest test has cases it fails, and this one has exactly one.

A straight line. If the bicycle was ridden dead straight, the two wheels leave the same track, and there is one mark rather than two. There is nothing to assign, no tangent that distinguishes anything, and no wheelbase to recover. The question has no content in that case rather than a wrong answer, and the test’s residual reflects it: both assignments come out perfect.

Everything else is decidable. A gentle curve is decidable, because the rear track’s curvature differs from the front’s; a circle is decidable, because the two tracks are concentric circles of different radii and only one of the two extension directions lands correctly. It was worth checking, because a pair of concentric circles is the case where an ambiguity might plausibly hide: run on a circle of 8 m with a 1.05 m wheelbase, the right assignment misses by 0.05 mm and the wrong one by 135 mm.

The failure case is therefore not a defect in the test; it is the observation that a bicycle ridden in a straight line has not left evidence of having a wheelbase at all.

The failure is a slope, not a cliff #

The one case the test cannot decide is the dead straight line, and stating it that way makes it sound like a knife-edge: everything decidable, and one measure-zero exception. It is not, and the honest form matters for anybody who would actually use this on a real pair of tracks.

The test works by comparing two misses — 0.13 mm the right way round against 447 mm the wrong way — and it is the ratio that carries the verdict. Both numbers depend on the path. The correct-assignment miss is set by how well the marks were measured and how finely they were sampled, and it does not care much about the shape. The wrong-assignment miss is set by how far the reverse construction is wrong, and that is set by curvature: a rear tangent extended forward lands on the front track either way when the two tracks nearly coincide, because there is nothing for it to miss by.

So as the path straightens, the numerator holds and the denominator collapses. The ratio of 3,554 on a well-wiggled path falls continuously toward one, and the test loses its power gradually rather than at a point. The straight line is where the ratio has reached one exactly; well before that, it has reached a value too small to be trusted against the measurement error in a set of tyre marks in mud.

That reframes the failure case usefully. The right question is not was the path straight but was it curved enough, and the answer is available from the marks themselves without knowing anything else: the separation between the two tracks is the observable, and a bicycle ridden straight has one track while a bicycle ridden on a curve has two whose separation grows with how hard it was turned. The test can therefore report its own confidence — compare the two misses and quote the ratio — which is a good deal better than declaring a verdict and a special case.

It is also the reason the wheelbase recovery has a sharp minimum rather than a flat valley. Sweeping the wheelbase and watching the residual is the same instrument read a third way, and the sharpness of the minimum is the same curvature dependence: a well-curved path gives a deep, narrow minimum and a nearly straight one gives a shallow basin in which many wheelbases fit almost equally. A reading that came back with a broad minimum would be saying this path does not determine a wheelbase, and that is information rather than a failure to converge.

There is a practical corollary worth having, since it costs nothing. A set of marks long enough to contain both a curved stretch and a straight one should be read on the curved stretch and the answer carried across, rather than averaged over the whole. Averaging dilutes a strong verdict with a stretch that had nothing to say, and the dilution is invisible in the mean — which is the same statistic-hiding-a-distribution problem that this site keeps running into, arriving here as a question about which part of the evidence to look at.

The general shape of it is worth naming because it is not special to bicycles. A geometric identification is only as strong as the geometry it was given. The relation between the two tracks is exact and holds on every path including the straight one; what varies is how much the wrong answer is punished, and that is a property of the evidence rather than of the method. A test with a perfect law and no discriminating power is a test that has been shown nothing, which is exactly what a straight line is.

Reading a mechanism from its output #

This essay is the only place on this site where a mechanism is identified from marks rather than measured while running, and the difference is worth drawing out because it changes what a check can be.

Everywhere else, the mechanism is available. A four-bar’s lengths are known and its motion is computed from them; the check is that a second computation agrees. Here the lengths are unknown and the motion is all there is, and the check has to be internal: the residual of a fit, and the ratio between the residual of the right hypothesis and the residual of a wrong one.

That ratio is doing the work that a second route does elsewhere, and it is worth being explicit that it is a weaker instrument. A residual ratio of 3,554 says the two hypotheses are not equally good; it does not say the winning hypothesis is right, only that it is enormously better than the specific alternative it was compared with. The way to strengthen it is to test more alternatives, and the alternatives worth testing are the ones that would produce nearly the same tracks: a longer bicycle, a shorter one, a tricycle with the same track spacing.

Sweeping the wheelbase is exactly that test, and it comes back with a single sharp minimum rather than a flat valley, which is the evidence that the family of alternatives really was explored rather than assumed away.

What would break it #

Three things would, and each is worth a sentence because each says what the test is actually assuming.

A skidding rear wheel. The whole construction rests on the rear wheel rolling. A rear wheel locked and sliding — a skid — leaves a mark that is not a rolling track, and the tangent test on it returns residuals of the wrong order in both directions rather than a clean verdict. That is a useful failure: the test reports that neither assignment works, which is itself information.

A wheelbase that changes. Nothing here has one, but a bicycle with rear suspension has a wheelbase that varies by a centimetre or two over its travel, and a fitted single value would come back with a residual larger than the sampling error and no obvious cause. The fit’s residual is the instrument that would say so.

Tracks that are not from the same vehicle. Two bicycles ridden along the same path leave two tracks with no relation between them, and the test would return large residuals both ways round. Again, it says so rather than silently picking a winner — which is the property a comparison of two hypotheses has and a single fitted number does not.

The general shape of all three: the test is a falsifiable reading of the marks. Its verdict comes with a residual, and a residual that is large on both hypotheses means the marks were not made the way the model says.

The same reading, on a mechanism with three tracks #

An articulated lorry leaves three tracks — front axle, tractor rear axle, trailer axle — and the same relation holds twice: the tractor’s rear track points at its front track at one distance, and the trailer’s track points at the kingpin at another.

That gives two independent identifications and one consistency check, which is more than a bicycle offers. It also gives a way to answer a question that gets asked in practice: whether a given set of marks could have been made by a given vehicle. Two rod lengths, both fitted, both with sharp minima, and both required to match a vehicle’s published dimensions.

Nothing in that is hypothetical machinery; it is the same one-line law applied twice, which is what a chain of towed axles is.

Why the towed wheel is the readable one #

One last asymmetry is worth naming because it explains why the test works at all.

A towed wheel’s heading is determined by where it has been and where the hitch is. It has no freedom of its own: it points along the rod, and the rod points at the hitch. A steered wheel’s heading is chosen, moment by moment, by whatever is steering it.

So the rear track carries the whole of the mechanism’s geometry and the front track carries the rider’s decisions. The rear track is a statement about the bicycle; the front track is a statement about the ride. The tangent test works in one direction and not the other because information flows in one direction and not the other, and a test that ignored that would be reading a decision as though it were a constraint.