The minor axis is not the axle

Every manual that shows how to draw a cart, a gun carriage or a spinning wheel gives the same instruction for the wheels. Draw the axle, then draw an ellipse with its short axis lying along it. The rule is compact, it is easy to execute with a straightedge, and it has the great advantage that the axle is a line the draughtsman has usually already drawn for other reasons.

The essay that measured the disc lying on the floor took two versions of that instruction — the short axis along the disc’s own axis, and the short axis pointing at the middle of the picture — and found the first out by three and a half degrees at the edge of an ordinary frame and the second out by seventy-nine. This one is about the standing wheel, where the rule takes a different and more seductive form, and about why the second rule is in circulation at all.

The seduction is that for a wheel the axle is not a direction on the paper. It is the image of a world line — the line through the wheel’s centre at right angles to the wheel’s plane — so it converges towards a vanishing point like everything else in the picture and leans differently in every part of the frame. A rule that says “put the short axis on that” feels like a projective statement rather than a rule of thumb. It is exactly true when the wheel’s centre lies on the principal ray, and false everywhere else, and the measurement below says by how much.

Where the rule is exactly true, and why it is exactly rather than nearly #

The zero on the middle wheel is worth explaining, because a rule that is exactly true somewhere is a rule with a reason and a rule that is nearly true everywhere is a coincidence.

Put the wheel’s centre on the principal ray. The arrangement — camera, wheel, axle — then has a mirror symmetry: reflect everything in the plane containing the optical axis and the axle, and nothing moves. The image is therefore symmetric about the line in which that plane cuts the picture, which is the axle’s image. An ellipse symmetric about a line has that line as one of its axes, and for a circle seen obliquely it is the short one. So the rule is not approximately right there; it is a theorem, and it holds to the last bits of a double.

Move the wheel sideways and the mirror plane no longer contains the optical axis, the symmetry is gone, and there is nothing left to make the two lines coincide. That is the whole mechanism, and it explains a property of the error that would otherwise be surprising: it is a function of the angle between the principal ray and the line to the wheel, and not of anything about the wheel.

It also explains why the rule is presented without qualification. A draughtsman setting up a drawing puts the interesting object near the middle of the sheet, and near the middle of the sheet the rule is very nearly a theorem. The wheels at the edge of a composition are usually the ones nobody is looking at.

It is worth saying what the exact object is, since the rule is an approximation to something rather than to nothing. The image of a circle lying in a plane is got by carrying the circle’s matrix through the plane-to-picture homography, which returns the image conic exactly and without sampling; its axes are then an eigenvector problem on a two-by-two block, and there is no fitting anywhere in the chain. The conic a circle becomes sets that machinery out. Everything measured below is a comparison between the axes of that conic and the drawn axle, so a disagreement is a fact about the two lines rather than a residual of a curve fit.

What it costs on the paper, which is much less than it sounds #

Five point nine five degrees sounds like a great deal for an error nobody notices. The reconciliation is that an ellipse’s orientation is a badly conditioned way of describing an ellipse when the ellipse is nearly round.

Rotating an ellipse by an angle $\theta$ moves a point on it by roughly $\theta$ times the difference between the two semi-axes, not by $\theta$ times the semi-major axis. A wheel seen at a moderate obliquity has axes that differ by a modest fraction, so a few degrees of misorientation is a fraction of a pixel of curve. The rounder the ellipse the worse the angle and the smaller the mark, which is a genuinely perverse combination: the error is largest in angle exactly where it is least visible.

That is the reason this belongs among the taught rules that are wrong rather than among the ones that are harmful. The rule produces a drawing that is out by a line’s width. A draughtsman who abandoned it in favour of computing the true axes would gain under a pixel and lose the only construction that can be done with a straightedge.

The measurement matters anyway, and for two reasons that have nothing to do with drawing carts. The first is that the rule is often stated as though it were a theorem, and reasoning from a false theorem does not stay bounded by the case that motivated it. The second is the subject of the next two sections: there is a rule of exactly the same shape that is a theorem, and the two are indistinguishable from any drawing.

A sphere obeys a rule of the same shape, and obeys it everywhere #

The other rule in circulation for a drawn circle says that its short axis points at the centre of the picture. On a disc it is out by seventy-nine degrees at the edge of a frame, which is nearly perpendicular to the truth. It is worth asking where such a rule came from, because a rule that wrong does not survive by accident.

That is a theorem and it is not a difficult one. A sphere’s outline in space is a circle, the cone of rays touching the sphere is a right circular cone with the line from the eye to the sphere’s centre as its axis, and the picture is a plane section of that cone. A plane section of a right circular cone is an ellipse whose axes are the section’s line of steepest slope and the horizontal square to it — and the line of steepest slope, projected into the picture, is the line towards the point where the cone’s axis would be perpendicular to the plane. That point is the principal point.

So a sphere’s picture points at the centre of the picture, exactly, everywhere, at any field of view. It is one of the very few unqualified statements about orientation that this collection can make, and the sphere’s stretching is where the familiar complaint about wide-angle photography comes from — which is a correct projection rather than a distortion and looks wrong for reasons the geometry accounts for completely.

The rule about circles pointing at the centre of vision is that theorem, applied to the wrong object. Somebody had the sphere’s behaviour in mind, or had seen it in a photograph, and generalised it to circles. The statement survives being written down because it is a true statement in the neighbouring case and because on the optical axis it agrees with the axle rule and with the truth.

The two rules on one axis #

The comparison is only worth anything if the two are measured against the same variable, at the same positions in the frame, by the same machinery. Otherwise a claim that one is exact and one is not is a claim about two different experiments.

The two lines do not merely differ in size. They differ in kind, and the shape of the plot says which is which without any need to read the numbers. One rises smoothly with the departure from the axis, which is the behaviour of an approximation whose neglected term is growing. The other is flat at a level twelve orders of magnitude below it, which is the behaviour of an identity being evaluated in floating point.

That is the discrimination this essay exists for, and it is worth being explicit that it is a discrimination rather than a debunking. Both objects are round. Both make ellipses. Both are drawn by the same projection through the same eye onto the same plane, and both rules are of the form “this ellipse’s axis lies along this line”. Nothing in a finished drawing distinguishes them. A draughtsman who has learned that ellipses point at things has learned something true, and has no way of learning which things.

There is one visible clue and it is not available in the ordinary case. A sphere’s ellipse points at the principal point, which for an untrimmed picture is the middle of the sheet — so on a print of known provenance the sphere rule is checkable with a ruler. A wheel’s axle points at a vanishing point, which is usually off the paper. The rule that can be checked is the one that is true, and the rule that cannot be checked is the one that has been repeated for five hundred years.

Why the rule survived, and it is not because draughtsmen were careless #

The strongest evidence that the axle rule is a good working rule rather than a bad one is what happens to it as the field of view narrows.

A quarter of a degree is nothing. On the narrow fields of view that panel painting, drawing-office practice and most photography before the twentieth century worked within, the axle rule is right to within the width of the pencil, everywhere on the sheet. It is not a rule that people got away with; it is a rule that was correct for the pictures they were making.

The band inside which the rule is safe is roughly the band the old advice about a cone of vision draws, and the coincidence is not one. The sixty-degree cone is a rule about how much angle a picture may subtend before objects near its edges start to look stretched, and stretching near the edge is the same quantity — the departure of the local map from a similarity — that makes the axle rule fail. The two are different consequences of one thing, which is that a plane picture cannot be locally similar to the scene except on the principal ray. So a draughtsman working inside the traditional cone was working inside the region where the rule holds to a fraction of a degree, and the rule and the cone were never independent pieces of advice.

What changes is the picture, not the geometry. A wide field of view puts the interesting part of a composition at a large angle off the principal ray, and that is where the rule goes. So the rule’s failure is a consequence of the same shift that produces every other wide-angle complaint — and the viewing distance printed on the same figure is the reason. A picture is correct from a distance proportional to its focal length, and a 90-degree picture shown 160 millimetres wide is correct from 8 centimetres, which nobody occupies. A rule of thumb that fails at 90 degrees across is failing in a picture that is already being read from the wrong place.

That is the honest account of how a false rule persists. Not carelessness, not a failure to check, but a rule tested exhaustively inside a range and then carried outside it by a change in the pictures rather than a change in the rule.

The centre goes wrong too, and it goes wrong first #

The axle rule fails in two ways at once and only one of them is about direction.

Five per cent of a width is a much larger mark on the paper than a few degrees of lean, so the ordering of the two faults is the opposite of the ordering of their notoriety. The essay that measured the centre makes the point that the gap is large enough to see and small enough to be dismissed as a slip; the lean is small enough to be invisible and is the one the manuals give a rule for.

The compounding also makes the wheel a worked example of a point the essay that got the centre back out of the picture makes in general — that a drawn conic carries enough information to recover the constructions its maker left out, so the corrections are available from the drawing itself rather than from the scene. That is the difference between a rule of thumb and a construction. A rule of thumb takes an input the draughtsman has and returns an answer; a construction takes the picture and returns the answer the picture already implies.

The two faults also have different fixes. The centre has an exact construction — the image of the circle’s centre is the pole of the plane’s vanishing line with respect to the image conic, so a straightedge and the horizon put a mark on it. The orientation does not have a comparable one-line construction, which is presumably why a rule of thumb grew in its place.

And they compound in the drawing rather than cancelling. The wheel drawn by the taught method has its ellipse centred on the wrong point and leaning the wrong way, and the two errors are independent — the axle line still passes through the drawn centre, so putting the ellipse’s middle there and its short axis along the axle produces a curve that is wrong in position and in orientation for two unrelated reasons.

What this does not settle #

The measurements are of a geometry and three things are outside them.

Nothing here says a drawn cart looks wrong. Under a pixel of curve is a geometric statement about which scene the drawing is a projection of, and whether any of it is visible is a question this collection has no instrument for. The rule’s survival is evidence about draughtsmen and their subjects rather than evidence about perception, and it is offered as the first.

The sphere result is about the outline of a sphere and does not extend to a round object that is not a sphere. An ellipsoid, a barrel or a head has an outline whose orientation depends on the object’s own axes as well as on the eye, and the exact statement then has the wheel’s character rather than the sphere’s. The clean theorem comes entirely from the rotational symmetry of the cone of tangent rays, and it goes as soon as that symmetry does.

And the exact ellipse computed here is the image of a mathematical circle. A real wheel has a rim of some thickness, spokes at some depth, and a hub that sticks out; its outline is not the image of a plane circle and does not have an exact ellipse to be compared against. The measurement is the right one for the rule as stated, which is a rule about a circle.

Two rules of the same shape #

The general form of this is the reason it is worth writing down, and it recurs across the collection wherever a rule is inherited rather than derived.

A rule of the shape “this drawn thing points at that other drawn thing” is checkable in one case and not in the other, is true in one case and not in the other, and gives the same instruction to a draughtsman in both. The sphere’s version is a theorem and reads at the arithmetic floor at every position in the frame; the wheel’s version is a first-order approximation about the principal ray and reads 9.39 degrees at the edge of a wide one. Nothing in the drawing tells them apart, and nothing in the manuals tries.

The same failure appears at least twice more here in different clothing. A construction that assumes the middle of the sheet is the middle of the picture is exact when the assumption holds and wrong with no residual when it does not, and the assumption is a fact about scissors. A grid cell filled in proportion is the true map on a wall square to the camera and a second-order approximation on a floor, and nothing in the method’s description mentions the plane.

In all three cases the rule is exact in a case that is easy to meet, common to learn on, and impossible to distinguish from the general case by looking at the result. The habit that catches them is the one this collection runs on: find the neighbouring case where the effect should be absent, and check that the number there is zero rather than small. A zero says a mechanism is missing. A small number says only that the day’s example was gentle.