The stair that turns has a vanishing point that moves

A straight flight of stairs is the tidiest object in a drawing manual, because every direction in it is one of four. The treads run across, the flight runs along, the risers run up, and the nosings run up the pitch — and the essay that counted them showed that the fourth is the interesting one, since nothing in the stair actually points along it and its vanishing point is off the horizon all the same. Four directions, four vanishing points, and a draughtsman who has found them has found everything the flight will ever ask for.

Turn the flight about a newel and that economy is gone. A spiral stair is one tread rotated by a constant angle at each step, so its treads run in as many directions as there are treads. Every one of those directions is horizontal, so every one of them has a vanishing point on the one eye-level horizon — and no two of the points are the same point.

The sequence of those points is a projectivity of the horizon, and four consecutive of them therefore have a cross-ratio that does not change as the stair climbs. That number is what this essay measures, and the reason it is worth measuring is that the formula behind it contains no focal length, no principal point and no eye. It is the turn per tread the builder chose, readable off a photograph nobody calibrated.

The twelve points are on one horizon, and only four are on the paper #

The first claim is the cheap one and it still has to be checked, because it is the claim the rest rests on. A tread’s front edge is horizontal. A horizontal direction lies in the ground plane. The ground plane’s vanishing line is the eye-level horizon, so the vanishing point of every tread edge is on it, whatever the stair does about a vertical axis. Twelve points, one line, and the residual against the camera’s own horizon runs at 2.8e-14 pixels.

That is a statement about the picture rather than about the stair. Rotating a direction about a vertical axis keeps it horizontal by construction, so a spiral stair contributes nothing at all off the horizon — which is precisely the difference from a straight flight, whose nosings point up the pitch and leave the horizon behind.

The awkward half is where the points actually are. Four of the twelve fall inside the paper. The rest are off it, the furthest by 2.22 canvas widths, and a draughtsman who wanted to run each tread edge to its own point would need a drawing board four or five times the width of the picture. That is the ordinary condition of a vanishing point rather than a defect of this stair — carrying a height across a room needs a point 3,300 canvas widths away — but it is the reason the sequence has to be read as a sequence rather than drawn.

Why the march is uneven #

The stair turns by a constant angle and the points do not move by a constant amount, and the reason is worth having plainly because it is the whole content of the word projectivity.

A direction in the ground plane, at bearing $\phi$ to the view, images at a vanishing point whose distance along the horizon from the principal point goes as $f\tan\phi$. Equal steps in $\phi$ are therefore not equal steps along the horizon: the tangent stretches near a right angle and compresses near zero. A tread whose edge comes close to parallel with the picture plane throws its vanishing point far out; a tread whose edge is nearly along the view brings it close in. The crowding at both ends of the walk above is that tangent, drawn.

The tangent also says where the sequence breaks the paper. As a tread edge comes parallel to the picture plane its bearing approaches a right angle, the tangent runs away, and the vanishing point leaves the drawing altogether and arrives at the horizon’s point at infinity. Nothing has gone wrong at that tread — the edge is still horizontal, its point is still on the horizon, and the horizon still has that point. What has gone wrong is the arithmetic of anyone holding the horizon as a Euclidean number line, which is why the sequence has to be read as a map of a projective line and not as a list of coordinates.

It is the same reciprocal that appears everywhere in this collection, and the essay on maps of a line states the general case in one sentence — turning a direction by a fixed angle is a projectivity of the horizon, applied once per turn. What that essay does not have is an object that performs it. A row of posts advanced by a bay is parabolic; a distance doubled is hyperbolic; a right angle is the involution. A spiral stair is the general elliptic case, built in oak.

The classification matters here rather than being ornamental. A parabolic map has one fixed point and everything crowds toward it; a hyperbolic map has two and points run from one to the other. An elliptic map has neither. There is no direction on the horizon that a rotation about a vertical axis leaves alone, so there is nowhere on the horizon for the sequence to accumulate, and it does not: it goes off one end and comes back from the other, indefinitely, which is what a line being a closed curve means in practice.

The measurement — four consecutive points read 1.353720 #

A projectivity of a line preserves cross-ratio, so four consecutive vanishing points have one, and it is the same four-mark quantity the whole collection is built on. For a turn of $\theta$ per tread it comes out in closed form:

$$\mathrm{CR}(\theta) = \frac{4\cos^{2}\theta}{1 + 2\cos 2\theta}$$

Nothing about the camera is in it, and the reason is worth spelling out because it is not luck. Four points on the horizon are the images of four horizontal directions, and the four rays from the eye through those points are parallel to those directions. A cross-ratio of four concurrent rays is a property of the pencil rather than of any line cutting it, so the number computed on the drawn horizon is the number computed on the pencil at the eye — and the pencil is fixed by the stair’s four bearings alone. The focal length decides where the four points fall; it cannot decide the ratio in which they divide each other. Moving the eye, changing the lens or cropping the print all move the four marks and leave the one number where it was.

Not the focal length, then, not the principal point, not where the eye stood, not how the picture was cropped. That is the point of the exercise: a reader with a photograph, a straightedge and four nosings has the same number as the builder, and inverting the formula gives the turn back.

Two features of the curve are worth separating. It is monotone, so the reading is invertible and a measured cross-ratio names one turn rather than two. And it approaches 4/3 as the turn goes to nothing, which is the value four equally spaced points always give and is therefore the value that carries no information — the tautology dividing depth by eye records, arriving here from a new direction.

The constancy is the part that had to be measured rather than asserted. Twelve treads offer nine overlapping quadruples of consecutive vanishing points, and if the sequence were merely regular-looking they would disagree. They agree to 3.1e-15 relative, which on a double is the arithmetic floor. So the number is a property of the stair and not of which four treads happened to be legible in the photograph.

Three treads fit the map, and nine test it #

Constancy of a cross-ratio is necessary for a projectivity and is not sufficient on its own, so the stronger claim is put to a stronger test. Three of the twelve vanishing points determine a map of the horizon to itself — three pairs fix a projectivity of a line, exactly as three pairs fix nothing less and nothing more. The map is then asked where the remaining nine go, and it has never been shown them.

The comparison is what makes the residual mean something. A number at 3.3e-11 pixels is only impressive against the alternatives, and the alternatives here are not straw men — an affinity and a translation are the two descriptions a draughtsman would actually write down for a sequence of marks along a line. They are out by 1116 and 449 pixels on a picture 690 pixels wide, which is to say they are not descriptions of this sequence at all.

The classification the fit returns is elliptic, and it is returned rather than assumed. That is the third of the three kinds of self-map a line admits, and it is the one with no real fixed point. The reason is geometric and short: a fixed point of the sequence would be a horizontal direction that the rotation sends to itself, and a rotation about a vertical axis by anything other than a straight angle sends no horizontal direction to itself. So the map cannot have one, and the eigenvalue arithmetic agrees.

The control — a straight flight has no cross-ratio to read #

Every measurement here should be run against the case in which the effect is absent, and for a turning stair that case is a stair that does not turn.

The twelve points collapse to one, to 0.0e+0 pixels, and that is the expected answer. The interesting part is what happens when the cross-ratio is asked for anyway.

The closed form tends to 4/3 as the turn goes to zero, so the obvious implementation returns 4/3 for a straight flight — smoothly, plausibly, and wrongly. Four coincident points have no cross-ratio: the quantity is a ratio of differences and every difference is nought. The routine refuses them. It is asked for a cross-ratio of four points that are one point and it declines rather than reporting the limit of a formula evaluated somewhere else.

That refusal is the check that stops the whole measurement from being decoration. A number returned at the degenerate input is a number that would also be returned by machinery that had computed nothing, and this collection has recorded before that a necessary condition tested where it cannot fail is not a test. The limit and the value at the limit are different objects here, and keeping them apart is the difference between a reading and a coincidence.

What a straight flight has instead #

The straight flight is not poorer than the spiral, only differently arranged, and the comparison is worth drawing because it locates exactly where the fourth direction went. This is the figure from the essay on the line every nosing is on, which measures a flight that does not turn.

So the two stairs answer the same question with opposite geometry. The straight flight has four directions and hides its dimensions in a single vanishing point off the horizon, at an angle that gives the rise over the run. The turning stair has twelve directions, all of them on the horizon, and hides its one dimension — the turn — in the spacing between them. Neither reading needs a ruler and neither needs a calibrated camera; both are angles and ratios recovered from marks.

The turning stair’s rise is a separate question with a separate answer, and the honest statement is that the cross-ratio does not touch it. Four vanishing points of horizontal directions know nothing about how far apart the treads are vertically. A spiral stair with the same plan and twice the rise draws the same twelve points in the same twelve places, so the turn is recoverable and the pitch is not, at least not from this construction.

The drawn stair is a shallow spiral, and a real one turns further #

The figures above turn 12° a tread, and a domestic spiral stair turns nearer 30°. That is not a detail to leave in a caption.

Twelve treads at 30° sweep 330°, and somewhere in that sweep a tread’s front edge comes parallel to the picture plane. Its vanishing point is then at infinity, which is a perfectly good point of the projective horizon and not a point the picture can draw or a least-squares fit can hold. Twelve treads therefore cap the turn at 16.4° — the constraint is that eleven steps stay under a straight angle — and the drawings here stop at 15°.

The measurement does not need twelve. A cross-ratio needs four, and the prediction test needs three to fit and any number to check. Four consecutive treads of a real 30° stair sweep 90°, every vanishing point stays finite, and the reading is the same reading. So the correct account of the pictures above is that they are a shallow spiral chosen so that the whole flight can be shown at once, and the correct account of the method is that it works on a real stair four treads at a time.

There is a second reason to prefer few treads on a photograph, and it is the ordinary one. A vanishing point far off the paper is located badly by a short bundle of nearly-parallel edges — the recovery essay prices that at a relative error growing with the distance — and the treads whose edges come nearest the picture plane are exactly the ones whose points fly out. Choosing four treads near the middle of the sweep is choosing the four best-conditioned points, and nothing about the invariant cares which four they are.

What the picture gives back, and what it does not #

The recovery is worth stating precisely because it is narrower than it first sounds and better than it looks.

What it gives. The turn per tread, in degrees, from four nosings in one photograph, with no calibration, no scene survey and no scale. The formula carries no camera, so a cropped picture, a scan of a print, or a frame from film all read the same. Inverting the measured cross-ratio returns 12.0000° against the 12 the stair was built at.

What it needs. That the four treads be consecutive, since the formula is about one turn repeated; that the horizon be identifiable, since the vanishing points are found by intersecting drawn edges and the reading is along that line; and that the stair really is a constant rotation rather than a set of treads laid out by eye. That last is a condition and not a check. A stair whose turns are unequal still yields four vanishing points and still yields a cross-ratio, and the number will name some constant-turn stair — the one whose four points sit that way.

What separates the two. More than four treads. Nine quadruples agreeing to 3.1e-15 is a statement no unequal stair can make, and a set of quadruples that disagree measures how unequal it is. This is the same structure the classification essay gives for a row of posts, where a trace close to four but not equal to it measures how far a row is from evenly spaced.

And how well it is conditioned. The curve is monotone but it is not steep. Between a stair that does not turn at all, where the formula tends to 4/3, and one turning 12° a tread, where it reads 1.353720, the cross-ratio has moved by about one part in sixty-six. A whole degree of turn near the bottom of the range is therefore worth a change in the fourth decimal place of a measured cross-ratio, and a cross-ratio measured off a photograph by clicking on nosings carries error in the third. So the reading is exact on a drawing and delicate on a print, and it is delicate in the direction that matters least — a shallow spiral is the case where the turn is hardest to read and also the case where it is nearest to a straight flight. The steepening of the curve with the turn means a real 30° stair is read far better than the 12° stair drawn here, which is the opposite of the usual arrangement in which the demonstration is the easy case.

And the whole of it says nothing about how a spiral stair looks, which is a question about seeing rather than about geometry. What is computed here is that the drawn points lie on one line, in an order fixed by a projectivity, at a spacing that names one angle.

One rotation, seen twice #

The finding generalises past stairs, and the generalisation is the reason the object is worth having.

A rotation of three-dimensional directions about a fixed axis induces a projectivity of the vanishing line of every plane perpendicular to that axis. The stair is the instance in which the rotation is furniture, and it is measurable because the picture happens to draw a dozen samples of the orbit. Any other object built as one shape rotated by a constant angle — a fan of louvres about a vertical pintle, the balusters of a curved balustrade, a spiral of paving setts — leaves the same signature on the same line.

That places this beside two other results in the collection rather than beside a manual’s chapter on staircases. The ramp’s own horizon says the eye-level horizon is one vanishing line among many, chosen by which plane the draughtsman stands on. The triangle of horizons says the same about the three coordinate planes at once. This essay says the horizon is not merely a line the drawing uses but a line with a group acting on it, and that the group’s orbit is legible in a photograph.

The practical residue is a single sentence. A photograph of four consecutive treads of a turning stair contains the angle the builder set out, exactly, and the reading survives everything a camera can do to a picture — because the only thing the reading uses is a ratio of four marks on a line, and that is the one quantity a projection keeps.