The line every nosing is on

A staircase is made entirely of level surfaces and vertical ones. The treads are horizontal, the risers are vertical, and there is nothing in it that slopes.

Its picture has a vanishing point that is not on the horizon.

Where it comes from #

The front top edge of step k sits one rise higher and one run further in than the front top edge of step k − 1. So the nosings — the front edges — all lie on one straight line in space, of direction (0, rise, −run): the line a plank laid on the flight would rest against.

That line is not a piece of the staircase. It is a relation between pieces. And like every other direction in a scene it has a vanishing point in the picture, which is where the vanishing point off the horizon comes from.

The word for this is worth being careful with. The ramp has its own horizon is about a sloping plane, and a sloping plane has a whole vanishing line of its own, above or below the true horizon depending on which way it tips. A stair is not a sloping plane; it is two families of surfaces neither of which slopes. What it has is a sloping line, and a line has a point rather than a line at infinity.

So the two cases are neighbours and not the same. A ramp beside the stair, at the same pitch, has a vanishing line, and the stair’s nosing point sits on it — which is a checkable incidence and is the reason the classical way to draw a flight is to draw the ramp first.

The point is not free #

Placing that point by eye is the obvious mistake, and it is not available, because the point is determined by things already in the picture.

The nosing direction, the direction of travel, and the vertical all lie in one plane of directions — they are three directions in the vertical plane that runs up the flight. Three coplanar directions have three vanishing points, and three vanishing points of coplanar directions are collinear in the picture.

So: find the vanishing point of the treads’ side edges (the direction of travel, on the horizon), find the vanishing point of the risers (the vertical), join them, and the nosing point is somewhere on that join. In a picture taken with the camera level the vertical point is at infinity straight up, and the join is the vertical line through the travel point — so the nosing point sits directly above it.

Measured, the three points here are collinear to within a part in ten billion of the area they would span if they were not.

A flight drawn with its nosings running to a point off that line does not depict a flight of stairs at all. It depicts something whose treads twist relative to its risers, and there is no staircase like that.

The pitch, read off the photograph #

The nosing point carries a number, and the number is the one a building inspector cares about.

The angle between two world directions is the angle between the rays their vanishing points sit on, which needs the focal length and the principal point and nothing else — and both of those are recoverable from a picture, which is the whole business of recovering the camera. Given them, the angle between the travel direction and the nosing direction is the pitch of the flight.

On the flight drawn here — 175 mm of rise to 280 mm of run — the angle read out of the two vanishing points is 32.0054°, and the angle the builder chose is 32.0054°. The rise-to-run comes back as 0.625000.

So a photograph of a staircase states its rise-to-run, and the only measurement made on the paper is the position of two points.

There is a tempting shortcut here that is wrong, and it was written first. The distance between the two vanishing points, divided by the focal length, is the rise-to-run — for a camera looking along the flight with its axis level. On the first camera that was not aimed along the flight it read a 0.625 stair as 0.698. A separation on the paper is a length in the picture plane; the ratio is an angle between rays; the two agree only where the rays are symmetric about the axis.

That is a small mistake with a general shape, and the shape is this collection’s most common one: the arithmetic was right and the objects were wrong. Both quantities are perfectly good quantities, both are computed correctly, and one of them is not the pitch.

What a flight drawn by eye depicts #

The construction is one thing; what happens without it is another.

Take the first nosing and the last where the projection puts them, and space the rest evenly down the picture between them. That is what a drawing does when it treats a flight as a repeating pattern — the steps are all the same, so draw them all the same.

Read back as heights, they are not the same at all.

A flight whose risers are 132 mm at the bottom and 229 mm at the top is not a flight anybody could walk up. Building regulations everywhere require the risers in a flight to be equal, and for a good reason: a person climbing stairs stops looking at them after two steps and relies on the rhythm. A step 100 mm out of pattern is how people fall down stairs.

So the by-eye drawing depicts a staircase that would be illegal to build and dangerous to use, and it looks entirely convincing, because a picture of a flight with steps that are equal on the page is exactly what a reader expects.

This is the same measurement as dividing depth by eye applied to a vertical sequence instead of a horizontal one, and it comes out the same way: an even spacing on the page is a geometric spacing in the room, and the two are only confusable because nobody reads the picture back.

How large the by-eye error is, before drawing it #

The spread from 132 mm to 229 is a measurement on one flight, and it has a form that lets a reader predict it on any other.

Spacing the nosings evenly down the page means making the image height linear in the step count. The true image height is a ratio of two linear functions of the count — one for the world height, one for the depth — so the correct drawing’s gaps are not equal, and forcing them equal redistributes the world heights in inverse proportion. The depicted rise at step $k$ is therefore proportional to the reciprocal of the gap the correct drawing would have put there, and

$$\frac{\text{largest depicted rise}}{\text{smallest}} \;=\; \frac{\text{largest correct gap}}{\text{smallest}}.$$

The by-eye flight’s error is exactly the perspective compression across the flight — a quantity already visible in a correct drawing of the same stair, and computable before anything is drawn. On this flight that ratio is 1.735, which is the 229 against 132.

For a flight running away from the camera the compression is close to the square of the depth ratio, since a drawn gap on a receding surface falls as $1/z^{2}$. So

$$\text{spread} \;\approx\; \left(\frac{z_{\text{near}}}{z_{\text{far}}}\right)^{2},$$

with $z_{\text{near}}$ and $z_{\text{far}}$ the distances to the bottom and top nosings. That is the whole of it, and it is unforgiving. A nine-step flight of 280 mm treads is 2.5 m deep; photographed from five metres, the depth ratio is 5 to 2.5 and the spread is a factor of four. Holding the spread under ten per cent needs a depth ratio under 1.05 — the flight photographed from twenty times its own depth, which for an ordinary domestic stair is fifty metres.

Two readings of that, and the second is the one worth carrying.

By eye is never safe on a flight. Not “usually inaccurate”: the arrangement in which it is acceptable is one nobody photographs or draws, because at fifty metres a staircase is a texture. The regulation the drawing violates — equal risers throughout — is violated by a factor of two in the ordinary case and by tens of per cent in the best case anybody would ever meet.

And the error is bounded by something a draughtsman can see. The spread is the ratio of the drawn gaps, so a draughtsman who has drawn even two steps correctly can read off how badly an even spacing will go wrong before committing to it. That is an unusually cheap check: it needs no camera, no focal length and no arithmetic beyond a ruler laid on two gaps.

It also says which flights are worst, and the answer is the ones that look most drawable. A flight running steeply away fills little of the page and has a large depth ratio; a flight seen nearly side-on has a depth ratio near one and is the case where an even spacing is nearly right — but a flight seen side-on is also the case where the nosing point is easy to find and the construction is trivial. The arrangement where the shortcut is safe is the arrangement where it is not needed, which is the same shape dividing depth by eye finds for a receding floor and the bay repeated by a straightedge answers with a construction that costs two lines a step.

How well the pitch comes out, and from what #

A number read off a photograph is worth having only with its tolerance, and the tolerance here is governed by two things that pull in opposite directions.

The nosing point is found by extending the line through the drawn nosings. A long flight gives a long line and a well-determined direction; a flight of three steps gives a short one, and a fraction of a millimetre of error at each end swings the direction noticeably. So more steps is better, and the improvement is roughly linear in the drawn length of the flight.

The travel point is found by extending the side edges of the treads. Those are short — one run each — and they are nearly parallel to the nosing line, which is the awkward part: the two lines whose crossing is wanted meet at the pitch angle, and for a shallow flight that is a shallow crossing. A stair at 32° is a comfortable case. A ramp at 6° is not, and the same drawing error costs five times as much.

The two together say something slightly surprising: the steeper the flight, the better the photograph measures it. A steep flight puts its nosing point close to the picture, well separated in angle from the travel point, and the crossing is clean. A shallow flight pushes the nosing point out toward the horizon, where it becomes hard to distinguish from the travel point at all — and in the limit of a level “flight”, which is a floor, there is no nosing direction and nothing to measure.

What else a staircase gives a photograph #

A flight of stairs in a picture is unusually generous, and it is worth saying what else it hands over.

It supplies three mutually perpendicular directions — the treads’ side edges, the treads’ front edges, and the risers — which is exactly what a camera calibration wants. Their three vanishing points make a triangle whose orthocentre is the principal point and whose shape gives the focal length. So a photograph with a staircase in it is a photograph that can be calibrated, without a calibration target and without knowing anything about the camera.

It supplies a repeated equal interval in two directions at once: equal runs going in, equal rises going up. That is the repetition a picture with nothing straight in it needs to find the horizon, and it is supplied twice over.

And it supplies a length, if the rise is known — and the rise of a staircase in a public building is one of the most reliably standardised dimensions there is. So the one thing a single picture cannot give, the overall scale, is available from a staircase in a way it is available from very little else.

The construction, which is short #

Given the horizon and the flight’s first step, the rest follows without any measurement:

1. The travel point. Extend the side edge of the first tread until it meets the horizon.
2. The nosing point. The nosing of the first step and the nosing of the last, joined, meet the vertical through the travel point there. Or, if the last step is not known, the pitch is chosen and the point is placed on that vertical at the height the chosen pitch requires.
3. Every nosing after that. The riser of step k rises from the tread of step k − 1 to the nosing line; the tread of step k runs back from that nosing to the next riser. Two intersections per step, both of them joins and meets.

Everything after step 2 is incidence, so the flight is exact for the same reason the constructions in carrying a height across the room and seven is not a power of two are exact: nothing in it quotes a length.

Step 2 is where a decision is made, and it is worth noticing what kind of decision it is. Placing the nosing point anywhere on the vertical through the travel point gives a perfectly good staircase — a different pitch, but a real one. Placing it off that line gives no staircase at all. So the point has one degree of freedom, which is the pitch, and the construction leaves exactly that free and nothing else.

Two flights, and the turn #

A flight that turns has two nosing points, one per flight, and they are not related in any simple way — the two flights run in different horizontal directions, so their travel points are different points on the horizon and their nosing points sit above different places.

What is related is the pitch: if the two flights are at the same pitch, which they usually are, then the angle each nosing point makes with its own travel point is the same angle. That is a check a drawing can be held to and a constraint a reconstruction can use.

A spiral stair has a nosing direction that changes continuously, so its nosings do not lie on a line at all — they lie on a helix, whose picture is a curve, and the whole construction here evaporates. The right way to think about that is that the straight flight is the case where a helix has infinite radius, and the vanishing point is what the curve’s asymptotic direction leaves behind.

What the stair is an instance of #

The general statement is worth pulling out because it applies to more than stairs.

A scene can contain a direction that no object in it points along. A stair is the clean example: nothing in it slopes, and the pitch is a real direction with a real vanishing point that carries a real measurement. A row of buildings of increasing height has a direction along their tops. A staircase of books on a shelf, a flight of terraces on a hillside, the corner of a stack of paper — all of them put a point in the picture that no edge of any object runs to.

Those directions are usually the ones worth measuring, because they are the ones that encode a relation — a rise per step, a rate of increase — rather than a dimension. And they are exactly the ones a drawing gets wrong, because a draughtsman draws the edges that are there.