A measuring point for a ramp

The measuring point is the construction that turns a perspective drawing from a picture into a measurement. Directions are easy — run a line to a vanishing point and the direction is exact. Distances along a receding line are not, and the measuring point is the extra point that settles them, by laying off true lengths on a line that runs across the picture and carrying each mark inward along a ray.

Every printed statement of the rule puts that point on the horizon, at the eye’s distance from the vanishing point of the direction being measured. That is not a convention and it is not an approximation. It is a correct statement about a plane, and the plane is the ground.

Move the construction onto a ramp and both halves of the rule have to move with it — the line the point stands on and the radius it stands at — and only one of them is ever mentioned. A ramp rising at 1 in 6.0 makes the difference plain: carried to the ramp’s own measuring point, six one-metre treads land where the camera puts them to 1.2e-13 pixels; carried to the ground’s, the sixth lands at 2.57 metres up the ramp instead of six.

The measuring point, restated so that it can be moved #

The construction has to be written in a form that says nothing about the ground before it can be carried anywhere else, and written that way it is three lines.

Let $\mathbf{d}$ be the receding direction being stepped along and $\mathbf{m}$ the direction of the measuring line, both unit vectors lying in the plane being measured. Stepping a distance $s$ along $\mathbf{d}$ and the same $s$ along $\mathbf{m}$ leaves an offset of $s(\mathbf{d}-\mathbf{m})$ — the same direction for every $s$, with only the length changing. So the segments joining the marks on the measuring line to the marks on the receding line are all parallel, and parallels meet at a vanishing point. The measuring point is the vanishing point of $\mathbf{d}-\mathbf{m}$, and that sentence contains no reference to the ground, the horizon or the level.

Two consequences fall straight out of it. Both $\mathbf{d}$ and $\mathbf{m}$ lie in the plane being measured, so their difference does too, and its vanishing point is on that plane’s vanishing line. And the distance from the measuring point to the vanishing point of $\mathbf{d}$ works out as $\sqrt{f^{2} + |V - \mathbf{p}|^{2}}$ — the eye’s own three-dimensional distance to the point $V$ on the picture plane, which is why the distance point is the viewing distance and why the radius depends on where $V$ is.

So the rule has two halves. The point stands on a line, which is the vanishing line of the plane being measured. It stands at a radius, which is the eye’s distance to that plane’s own vanishing point. On the ground both come out as the familiar recipe, because the ground’s vanishing line is the eye-level horizon and the ground’s vanishing point is on it. On a ramp neither does.

The radius is the half that is easiest to get wrong while believing it has been got right, because the recipe most often printed for it names the wrong point to measure from. The radius is the eye’s distance to $V$, the vanishing point of the direction being stepped along — a slant length, $\sqrt{f^{2} + |V - \mathbf{p}|^{2}}$, with the focal length as one leg of a right triangle and the drawn offset of $V$ from the principal point as the other. It is not the focal length, and it is not the eye’s distance to the centre of the picture.

Those three quantities coincide in exactly one arrangement, and it is the arrangement every textbook diagram is drawn in. Point the camera straight down the receding direction and $V$ lands on the principal point: the offset is nought, the slant length collapses to $f$, and the radius is the viewing distance itself. That is a one-point view of a pavement, it is where the distance point gets its name, and it is the second place in this construction where the demonstration case quietly stands in for the general one.

The figures here are drawn that way on the ground and not on the ramp, which is what makes the two radii comparable. The ground’s receding direction runs straight down the view, so the ground’s radius is the focal length itself and comes out at 707 pixels on this 52-degree picture. The ramp’s receding direction rises, so its vanishing point is not on the horizon at all — it stands 118 pixels above it — and the ramp’s radius is the hypotenuse of those two: 707 and 118 give 717, which is what the construction reads. The 1.38 per cent is that right triangle and nothing else, and it is small for the ordinary reason a hypotenuse is close to its longer leg.

Both halves move, and they do not move equally #

The two halves can be separated, because the wrong construction can be built halfway: the ground’s radius placed on the ramp’s own line. That is not a construction anybody would defend, but it is exactly the diagnostic — it holds one mistake fixed and removes the other.

Read those two numbers against each other and the diagnosis is immediate. The radius is almost right and the line is not. A 1.38 per cent change in a radius of 707 pixels is ten pixels; the line the point stands on moves 118. A draughtsman who carefully recomputed the distance-point radius for the ramp, congratulated himself on his rigour, and left the point on the horizon would have corrected the smaller half of the error by an order of magnitude.

The 118 pixels is not an arbitrary number either. It is $f\tan\theta$ for the ramp’s slope — the same quantity that separates the ground’s vanishing line from the ramp’s, which the essay that found the ramp’s own horizon established. The measuring point rides on its plane’s vanishing line, so it is displaced by exactly the amount that line is displaced, and the two facts are one fact met twice.

The wrong radius costs 0.083 metres and the wrong line costs the other 3.35 #

The halfway construction is worth dwelling on, because it converts a comparison of pixels into a division of blame, and the two are not the same argument.

Ten pixels against a hundred and eighteen says the radius is the smaller mistake as a displacement of the measuring point. It does not follow that the radius is the smaller mistake in metres of ramp, and it did not have to: a construction is not linear in the position of its measuring point, and a small displacement in the wrong direction can cost more than a large one in a harmless direction. The halfway construction is what settles it. Placing the ground’s radius on the ramp’s own vanishing line keeps the second mistake and removes the first, and it puts the sixth tread 0.083 metres out.

So of the 3.43 metres the taught rule loses, the wrong radius accounts for 0.083 and the wrong line accounts for the remaining 3.35 — a ratio of about forty to one. The construction is almost completely insensitive to the radius and almost completely determined by the line. That is a stronger statement than the pixel comparison and it points in a practical direction: an unmeasured drawing whose measuring point sits on the correct line at a radius guessed by eye will step off a ramp better than a carefully computed radius sitting on the horizon.

The reason is visible in the construction rather than in the arithmetic. Sliding the measuring point along a line changes the pitch of the carrying rays and leaves the family of them intersecting the same receding line; lifting it off the line changes which line the rays are pitched about, and the marks they cut are then cut on a receding line the rays were never aimed at. One is a change of parameter within a correct construction and the other is a construction of something else.

This also decides what a draughtsman working from a printed rule should be told first. Not recompute the distance point for the ramp — that is the forty-first part of the correction. Move the point onto the ramp’s own vanishing line, and the drawing is right to within eight centimetres in six metres before the radius has been touched at all.

The measurement — six metres stepped off lands at 2.57 #

The pixels have to be turned back into metres before the size of the mistake means anything, because a hundred pixels near a vanishing point is worth a great deal more ramp than a hundred pixels near the foot.

Three things in that plot are worth taking separately.

The correct construction is exact, not close. A residual of 1.2e-13 pixels is the arithmetic floor of a double, which is what it should be: the measuring point is a theorem rather than a device, and a theorem implemented correctly has no error term. This is the same standard the collection holds every construction to — the three-point layout places every corner to three parts in ten million million of a pixel — and it is what makes a residual anywhere above the floor evidence of something rather than noise.

The error is not a scale factor. The ground’s curve falls away from the identity immediately and keeps falling, so a draughtsman cannot rescue the drawing by shrinking the whole flight. The first tread is nearly right and the sixth is 3.43 metres short of six; the marks are not a mis-scaled ramp but a differently-shaped one, crowding toward a vanishing point that is not the ramp’s.

And the error is in the direction that flatters the drawing. The treads come out closer together than they should, so the flight looks shorter and steeper rather than broken. Nothing in the picture announces itself as wrong. That is the shape of every entry in this collection’s catalogue of taught rules — the rule that draws another room produces a perfectly good picture of a room nobody specified — and it is the reason the errors survive in print.

The control — a level ramp, and why nobody discovers this laying a pavement #

The measurement is only worth anything if the case it claims to distinguish actually reads differently, so the ramp is flattened and the same two constructions are run again.

That is the whole social history of the mistake in one figure. A rule tested on the case it was derived for cannot fail, and the ground is the case every measuring-point construction is derived for and demonstrated on. Alberti’s pavement is level. A tiled floor is level. The bay repeated along a colonnade is level, and the straightedge that repeats it never leaves the ground plane. A draughtsman can spend a career setting out pavements and courtyards and colonnades and never encounter an input on which the taught rule and the correct rule differ by anything at all.

This collection has a name for that situation. A necessary condition evaluated at the one input where it cannot fail is not a test, and a rule verified only where the two candidate rules coincide is not a verified rule. The ramp is the input that separates them, and it separates them by 3.43 metres.

How fast it grows, and where it stops growing #

One number at one slope is an anecdote, so the top tread’s error is swept across the whole range of ramps a building actually contains.

The curve rises fastest where ramps actually are. A wheelchair ramp at 1 in 20, a loading bay at 1 in 10, a garage ramp at 1 in 6, a stepped street at 1 in 4 — the whole of practice sits on the steep first third of that curve, and there is no shallow regime in which the taught rule is merely a little wrong. At one in six, six one-metre treads have become 2.57 metres of ramp: the drawing has lost more than half of what it was asked to lay off.

The flattening at the top is not the error stopping. It is the construction running out of ramp to be wrong about, because the marks it places have crowded up against the ramp’s vanishing point and there is a bounded amount of ramp between the foot and that point. An error that saturates because the drawing is full is not an error under control.

What the ramp’s own horizon already said #

The result here is a second consequence of a fact this collection established three rungs down, and the figure that established it is worth putting beside the ones above. This is from the essay that found the ramp’s vanishing line.

The relation between the two essays is worth stating plainly, because it is the difference between a curiosity and a method. That essay showed the ramp’s vanishing line exists and can be read off a photograph. This one shows what it is for: it is not merely a line the uphill edges happen to meet on, it is the line every metric construction on the ramp is anchored to. The vanishing line of a plane is the plane’s own horizon in the full working sense — the thing the draughtsman would call the horizon if they were standing on that plane rather than on the ground.

Which is the same claim the triangle of horizons makes about the three coordinate planes at once, arrived at from the metric side. Nothing distinguishes the eye-level horizon except that the ground is where things stand.

The condition nobody prints — the camera has no yaw #

There is a precondition on the whole construction, on the ground as much as on a ramp, and it is worth extracting because no printed statement of the measuring point mentions it.

A measuring line carries true lengths only when it is parallel to the picture plane. That is what makes the marks on it uniformly spaced, and it is the entire reason a ruler may be used on it directly. The across-the-view direction is parallel to the picture plane exactly when the camera has no yaw — when the optical axis has not been swung to one side. Yaw the camera and the across-direction acquires a finite vanishing point, the marks on the measuring line stop being evenly spaced, and every distance laid off along it is wrong before any measuring point is chosen.

The camera in every figure here is level and unyawed on purpose, and stating that is not modesty about the demonstration. It is the standing assumption behind every pavement construction in the literature, unstated, and it is the reason a construction that works on the drawing board can fail on a photograph that was taken with the camera turned. A tilt is separable and repairable — straightening a photograph does not move the eye — but the repair has to be made before the measuring line is trusted, not after.

What this does not settle #

The construction is exact and its reach is narrow, and the boundary is worth drawing.

It measures along the ramp, in the ramp’s own plane, in the ramp’s own metres. It says nothing about the vertical rise those treads correspond to, which is a separate question answered by a separate direction. It says nothing about how the ramp meets the ground at the foot, where the two planes’ constructions have to agree on one line and do — that line is the planes’ intersection, and it is fixed pointwise by both.

It also says nothing about finding the ramp’s vanishing line on a photograph nobody drew. Every number here comes from a camera whose focal length is known, so the ramp’s line is computed rather than fitted. On a real photograph the line has to come from the ramp’s own uphill edges, and it is then as well or badly determined as those edges are long and as their bundle is spread — the ordinary conditioning that the recovery essay prices. A ramp photographed nearly end-on gives a poorly located vanishing point and a measuring point with a large uncertainty riding on it, and the construction inherits all of it.

And it says nothing about whether the mistake matters to a viewer. A flight drawn 2.57 metres long instead of six is a perfectly plausible flight of a different building, and nothing about looking at it reveals which. That is why the error is measured against the camera rather than against an opinion.

One rule, one plane at a time #

The finding is small and its generalisation is not.

Every metric construction in perspective is a construction on a plane, and every one of them is written in the literature with the ground substituted for the plane, because the ground is where the demonstrations were drawn. The measuring point is the clearest case because it has two halves and both of them have to move. But the pattern is the same for the distance point, for the diagonal that finds a midpoint, for the division of a receding line into equal parts, and for the transfer of a height from one place to another: each is anchored to a vanishing line and a radius, and each is taught with the ground’s.

The correction is not a new construction. It is the old one with the words the horizon replaced by this plane’s vanishing line and the words the distance point replaced by the eye’s distance to this plane’s vanishing point. On the ground nothing changes, which is why the substitution survived. On a ramp at one in six it is worth 3.43 metres in six, and on a flight of stairs whose nosings have their own line it is worth the difference between a stair that could be built and one that could not.