Visual Plane
every picture states the point it is correct from
Constructing a view

The line every nosing is on

Nothing in a staircase points up the pitch. Every surface in it is level or vertical, and its picture has a vanishing point off the horizon all the same — belonging to the line the front edges of the treads lie on. That point is not free: it is collinear with the travel point and the vertical point, and the angle it makes says the rise-to-run the builder chose.
17 min read 13 figures Projected, not constructedOne picture, one lengthTaught, and never measured

Worth reading first: The ramp has its own horizon · One, two and three point are one construction · Where parallel lines meet.

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.

The line every nosing is onNothing in a staircase points up the pitch — every surface is level or vertical — and the picture has a vanishing point off the horizon anyway. It belongs to the nosings, it sits 32.01° from the direction of travel, and that angle is the rise-to-run the builder chose.horizoncorrect from 20 cm, at 160 mm wide44° across
Fig. 1 A flight of nine steps, with the line the front edges of the treads lie on continued to its vanishing point. Nothing in the staircase runs along that direction — every face of it is level or upright — and the point is a long way above the horizon all the same.
The line every nosing is onNothing in a staircase points up the pitch — every surface is level or vertical — and the picture has a vanishing point off the horizon anyway. It belongs to the nosings, it sits 25.11° from the direction of travel, and that angle is the rise-to-run the builder chose.horizoncorrect from 20 cm, at 160 mm wide44° across
Fig. 2 A shallower flight. The nosing point moves down toward the horizon as the pitch decreases, and in the limit of a level floor it merges with the travel point and there is nothing to measure.

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.

The three vanishing points of one box, drawn to scale with the boxThe picture is the small rectangle. Two of the three vanishing points fall well outside it, which is why they are computed rather than located by eye.orthocentrethe pictureVP₁VP₂VP₃focal length from the triangle — 853.9 pxspread 1e-14% across three routes
Fig. 3 The general fact this depends on. Vanishing points of directions lying in one plane are collinear in the picture, which is what puts the horizon where it is and what pins the nosing point to a line through two points already drawn.

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.

A box drawn from a known camera, and the camera recovered from the drawingThree vanishing points found from the twelve drawn edges alone give back the focal length to 4e-15 relative.recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across
Fig. 4 Where the focal length comes from. Three mutually perpendicular directions in the scene put their vanishing points at the corners of a triangle whose orthocentre is the picture’s centre, and the focal length falls out. A staircase in a building supplies all three for free.

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.

The same two ends, the steps put in by eyeThe first and last nosings are where the projection puts them and the rest are spaced evenly between. Read back as heights the rises run 132 mm to 229 mm.horizoncorrect from 20 cm, at 160 mm wide44° across
Fig. 5 The same two end steps, with the seven between them spaced evenly down the picture. The large marks are where the projection puts the nosings; the small ones are where the even spacing puts them. The two agree at the ends, by construction, and nowhere else.
The rises a by-eye flight depictsNosings spaced evenly down the picture between the same first and last step, read back as heights: 132 mm at one end and 229 mm at the other, a spread of 1.74. The flight drawn is one nobody could walk up.step 1 to 2132 mmstep 2 to 3142 mmstep 3 to 4152 mmstep 4 to 5164 mmstep 5 to 6178 mmstep 6 to 7193 mmstep 7 to 8210 mmstep 8 to 9229 mmthe flight as built: 175 mm, every stepevery step the same on the paperand not in the room
Fig. 6 The rises the even spacing depicts, read back as world heights. They run from about 132 mm at the bottom of the flight to about 229 mm at the top — a spread of nearly a factor of two, in a flight drawn from a first and last step that are exactly right.

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.

Two versions of the same invariant, one of which measures nothingFour consecutive divisions give the equal-steps method a perfect score. Using the vanishing point as the fourth point rejects it by 14%.error against the value the projection must producefour divisionsthree plus the VPthe projectionexactexactequal stepsexact14%halving3.6%25%tapering0.9%21%green: agrees with the projectiona necessary condition is not a test
Fig. 7 The general form of the failure. Four equally spaced things in the room have a cross-ratio in the picture that four equally spaced things on the page do not, and the test is available to anybody with a ruler and no knowledge of the camera.

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 one pixel of click error costs, against distanceA 1.75 m object at 3 m is measured to 0.29% per pixel; the same object at 201 m to 19.0% per pixel. The whole object shrinks toward the horizon, so a pixel buys more world.0510152050100150200distance from the camera (metres)error in the recovered height, per pixel of click error (%)5 m — 0.48%25 m — 2.37%100 m — 9.45%190 m — 17.94%one pixel, on a 690 px picturelinear in distance
Fig. 8 The general shape of the accounting, drawn for the single-view height measurement rather than for the pitch. A pixel of error in the drawn positions costs a certain amount in the recovered quantity, and how much depends entirely on how squarely the lines whose crossing is being taken meet.
The rises a by-eye flight depictsNosings spaced evenly down the picture between the same first and last step, read back as heights: 142 mm at one end and 214 mm at the other, a spread of 1.51. The flight drawn is one nobody could walk up.step 1 to 2142 mmstep 2 to 3153 mmstep 3 to 4166 mmstep 4 to 5180 mmstep 5 to 6196 mmstep 6 to 7214 mmthe flight as built: 175 mm, every stepevery step the same on the paperand not in the room
Fig. 9 The same by-eye spacing over a shorter flight. Fewer steps, the same spread of depicted rises, and the same reason: an even spacing on the page is a geometric spacing in the room.

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.

A 3.4 m object measured from one picture, 8 m awayThe base, the horizon crossing, the top and the vertical vanishing point have a cross-ratio of 1.9101. With the eye at 1.62 m that gives 3.400 m, against a true 3.4 m. The camera is not consulted.horizon — the eye's own heighthorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.3 cm per pixel of click error
Fig. 10 What the scale unlocks. Once one real length in the scene is known, every other length on the same plane follows, and a staircase’s rise is a length that is very nearly guaranteed.

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.

The same cube turned 30° — a three-point constructionNothing about the construction changed. The number of vanishing points inside any finite distance is 3, and 1 of them fall on the canvas.horizon3 vanishing points at a finite distance1979 px · 8129 px · 676 px
Fig. 11 The neighbouring fact about vanishing points. One, two and three point perspective are one construction with the box turned, and the number of points is a fact about the orientation rather than about the method. A staircase adds a fourth point to whichever of those the building already has.

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.

The ramp's own horizonEvery plane has a vanishing line, and a ramp's is not the ground's. The ground's horizon is the dashed line; the ramp's vanishing line is the one above it, and the ramp's uphill edges meet on it. The angle at the eye between that meeting point and the vanishing point of the same direction taken level is the gradient: 22.0000° recovered against 22° built, out of the picture alone.nothing on the ramp images above its vanishing linehorizonuphilllevelcorrect from 10 cm, at 160 mm wideslope recovered 22.0000° against 22° built
Fig. 12 The case next door, and the one the stair is usually confused with. A sloping plane has a vanishing line of its own, displaced from the true horizon by the slope. The stair’s nosing point lies on the vanishing line the ramp under it would have, which is the incidence that ties the two together.
The plan and the picture, drawn from one cameraThe rays in the plan and the edges in the picture are the same projection seen from two directions.plan, looking downpicture planeone camera, two views of it40° across
Fig. 13 The other way to get all of it, kept here for contrast. Draw the plan, draw the elevation, project. It is exact and it needs the plan, and the whole appeal of the nosing point is that it needs only the picture.

Where this goes next

nothing on the ramp images above its vanishing linehorizonuphilllevelcorrect from 10 cm, at 160 mm wideslope recovered 22.0000° against 22° built The ramp has its own horizon Every plane has a vanishing line, and a ramp's is not the ground's. Its uphill edges meet on a line above the horizon, and the angle at the eye between that meeting point and the same direction taken level is the gradient — 22.0000° recovered against 22° built, out of the picture alone, with no scale, no ruler and nothing known about the scene except that the ground is level. horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px One, two and three point are one construction The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity. horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon11.72 mhalve the remaining gap225.13 mtaper by eye347.80 mcorrect from 26 cm, at 160 mm wide34° across Dividing depth by eye Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

CollinearityFocal lengthForeshorteningHorizonIncidenceInclined planeOrthocentresingle-view metrologyStraightedge constructionVanishing point