Both vanishing points on the paper

Every book on two-point perspective begins by drawing a horizon and putting a vanishing point near each end of it. It is presented as a matter of arrangement — spread them out and the box is squat, bring them in and it is dramatic — and the advice is usually to keep both of them on the sheet so the construction lines can be drawn.

That last instruction is not about the sheet. It is about how far away the reader has to stand.

Why the separation is the focal length #

Two horizontal directions at right angles have vanishing points, and those two points and the principal point are tied together by the same relation that fixes the third point in the third point put where it looks right: the rays through the two vanishing points must be perpendicular.

Written out, with the principal point midway between the two, that condition says the focal length is half their separation.

There is no room in it. The focal length is not something a two-point drawing is free to choose after placing its points; it is what the placement is. And the focal length, scaled to the width the picture is displayed at, is the distance the reader’s eye must be from the paper. That identity is this collection’s founding claim and is set out in the point to stand at.

So a layout with the two points at the two edges of a page — one page-width apart — has a focal length of half a page, which is a 90° field of view, which printed at 160 mm is a picture correct from 80 mm.

Eighty millimetres is a hand’s breadth. Nobody reads a book at 80 mm.

What the reader gets instead #

A picture read from further away than it is correct from does not look wrong. It looks like a picture of a different room.

Standing in the wrong place measures the substitution exactly: reading a picture from k times its correct distance depicts a scene k times as deep, with every mark on the paper unmoved. Nothing about the drawing changes; what changes is what it is a drawing of.

At 400 mm — an ordinary reading distance for a book — a picture correct from 80 mm is being read at five times its distance, and depicts a room five times as deep as the one the draughtsman laid out. A hallway becomes a corridor. A room becomes a hall.

The line has no knee and no threshold, so there is no reading distance at which the substitution starts. It is worth seeing the same picture at a second distance for that reason alone: doubling how far back the reader sits doubles the depth of the room the picture depicts, and changes nothing else about it at all.

The table #

Working through the layouts a book might use, printed at 160 mm and read at 400:

The row a reader wants is the fourth from the bottom: four and a half page-widths apart, which puts both vanishing points comfortably off the drawing board, is the layout that is honest at arm’s length.

That is the whole of the finding, and it is uncomfortable in a specific way. The advice to keep both points on the sheet is not an aesthetic preference that happens to have a geometric consequence. It is an instruction to draw a very wide-angle picture, given to people who are drawing pictures that will be looked at from a normal distance.

The three-metre corridor #

It is worth working one case all the way through, because the abstraction hides how large the substitution is.

A draughtsman lays out a corridor. It is three metres wide, three metres high and ten metres long, and it recedes from the reader. Both vanishing points go near the edges of a sheet 160 mm across, one page-width apart.

That layout is a 90° lens. The drawing is a correct picture of the corridor from 80 mm in front of the paper — from there, the corridor is ten metres long and everything is where it should be.

A reader holds the sheet at 400 mm and sees a corridor fifty metres long, three metres wide and three high. Not a corridor at all: a tunnel. The far end, which the draughtsman drew as a doorway ten metres off, is now a doorway fifty metres off, and the reader’s sense of the building is wrong by a factor of five in one direction and correct in the other two.

That anisotropy is the thing to hold on to. The substitution is not a scaling, which would be harmless — a picture of a model is a picture of the thing. It is a stretch along one axis only, so proportions change: the corridor’s length-to-width goes from 3.3 to 16.7. A square room becomes a long one. A cube becomes a box.

And the reader has no way to know. Every alternative reading of the drawing is internally consistent; the picture is a perfectly good picture of the fifty-metre tunnel, taken from 400 mm, and nothing in it prefers the corridor.

Where the number is, on the paper #

For a reader who wants to check a published drawing, the whole calculation is two measurements and a division.

Measure the distance from the middle of the picture to one vanishing point, and to the other; call them a and b, taken as distances along the horizon on either side of the picture’s centre. The focal length in the same units is the square root of a times b — that is the perpendicularity condition, written for the general case where the two points are not symmetric.

Then divide by the picture’s printed width and multiply by it again in millimetres: the correct viewing distance in millimetres is the printed width times the square root of ab, divided by the width in the same units the points were measured in.

For the symmetric case a equals b and the square root of ab is half the separation, which is the rule quoted at the top. For an asymmetric layout it is the geometric mean, which is smaller than the average — so an asymmetric arrangement is always a wider picture than its separation alone suggests.

Two measurements and a square root, and the answer is a physical distance a reader can hold the page at. There is very little else in the theory of pictures that a person can check with a school ruler in half a minute.

What makes it checkable is that both quantities live on the paper. The separation of the vanishing points is drawn; the printed width is drawn; and the thing they produce between them is a distance in the reader’s own room. No property of the depicted scene enters anywhere, which is why the answer is available from a photocopy of a drawing whose subject is unrecognisable.

Where the advice comes from #

It comes from the drawing board, and it made sense there.

A construction with a vanishing point off the paper is a construction that cannot be carried out with a straightedge, and the repairs are awkward: a beam compass, a long batten pinned to the desk, an auxiliary construction that produces the converging lines without ever reaching their meeting point. All of them are more work than putting the points on the sheet, and all of them were normal practice in a drawing office precisely because the alternative produces a distorted picture.

So the instruction that reached the general-interest books is the shortcut, and the reason for not taking it stayed in the trade.

The same thing is measured in this collection from the other end. The height-transfer construction in carrying a height across the room is exact and its one intersection lands thousands of canvas widths off the page about half the time; that is not a defect of the construction, it is the ordinary situation, and a draughtsman deals with it. Vanishing points are usually off the paper. Every method that assumes otherwise has quietly assumed a wide lens.

What a wide layout does to the picture itself #

The depth exaggeration is what the reader gets. There is a second consequence, which is what the drawing looks like even before anybody stands anywhere.

A 90° field of view puts the corners of the frame 45° off the optical axis, and off-axis a sphere images as a noticeably elongated ellipse, a face is broadened, and a cube near the edge of the frame is sheared. Those are not errors — a correct projection does all of them, and they disappear when the picture is read from its own station point. But read from further away they do not disappear, and they are what people mean when they say a wide picture “distorts”.

This is the same subject as wide-angle is not distortion, reached from the drawing board instead of from the camera. A draughtsman who puts both vanishing points on the sheet has chosen a 90° lens without owning one.

The rule that was already there #

There is a piece of traditional advice that points the same way and is usually explained badly: keep the subject inside a 60° cone of vision.

The sixty-degree cone of vision measures the reason usually given — marginal stretch — and finds it to be exactly zero from the station point, so the rule is not about the picture at all. It is a statement about the reader: a picture whose field of view is 60° is correct from a distance a bit less than twice its width, which is about where a person holds a book.

Expressed as a vanishing-point separation, a 60° field is about 1.7 page-widths. So the cone rule and the both-points-on-the-page rule are direct contradictions of each other, given in the same books, usually within a few pages.

The cone rule is the one that is right, and it is right for a reason its usual defence does not mention.

Asymmetric layouts, and the hidden crop #

One more thing the separation decides, which is easy to miss.

The relation between the focal length and the two points assumes the principal point is midway between them. It usually is not: a composition with one vanishing point near the middle of the sheet and the other far off to the side is a common and perfectly reasonable arrangement.

When the two are not symmetric about the middle of the frame, one of two things is true. Either the principal point is not in the middle of the frame — the picture is a crop of a wider one, or was taken with a shifted lens — or the two directions are not perpendicular, and the box being drawn is not rectangular.

Both are real cases. What is not a case is “the principal point is in the middle, the box is rectangular, and the two points are wherever they were put”. The drawing has to give up one of the three.

That is the same fork as in the previous essay and it is worth noticing that it appears here too, because it is where most real layouts sit. A drawing is not usually wrong by having its points in absurd places; it is wrong by having a slightly asymmetric arrangement and no view about which of the three assumptions it is spending.

What asymmetry costs, in numbers #

The geometric mean is introduced above as the general case and left there. Working it out changes the table, because almost no real layout is symmetric and the asymmetry always runs the same way.

Hold the separation fixed at one page width and let the principal point sit off centre, so the two points are at distances $a$ and $b$ with $a + b = W$. The focal length is $\sqrt{ab}$, and

$$\frac{\sqrt{ab}}{(a+b)/2} = \frac{2\sqrt{r}}{1+r}, \qquad r = b/a,$$

which is 1 at $r = 1$ and falls away on both sides. At $r = 2$ it is 0.943, at $r = 4$ it is 0.800, at $r = 9$ it is 0.600, at $r = 16$ it is 0.471.

Put that back into the table’s own arrangement — both points on a 160 mm page:

Every asymmetric layout is a wider picture than its separation suggests, and never a narrower one, because the geometric mean of two positive numbers is at most their arithmetic mean. That is worth stating as a direction rather than a magnitude: whatever a reader estimates from the separation alone, the true field is wider and the true reading distance shorter, so the estimate errs on the comfortable side every time. A rule of thumb that is wrong in a known direction is a good deal more use than one that is merely approximate, since it can be applied and then corrected rather than applied and then doubted. So a composition described as “dramatic” — one point near the edge of the sheet and the other well off to the side, which is the arrangement most books recommend for a lively box — is not merely on the 90° row of the table. It is somewhere below it, and a three-to-one arrangement inside one page is a 118° picture correct from a distance at which nothing can be focused.

The limit of that is worth naming, because it is a hard constraint rather than a trend. As one vanishing point approaches the principal point — the middle of the frame, on an uncropped drawing — $a$ goes to zero and so does $\sqrt{ab}$. A zero focal length is not a picture. What the arithmetic is saying is that two perpendicular horizontal directions cannot have one of their vanishing points at the principal point and the other anywhere finite: if one direction runs along the optical axis, the perpendicular one is parallel to the picture plane and its vanishing point is at infinity. That is a one-point picture, and it has one point on the page by geometric necessity rather than by choice.

So a layout with one vanishing point near the middle of the sheet and another on the sheet is not a wide two-point picture. It is not a two-point picture of a rectangular box at all, and the fork the section above sets out has already been taken: either the box is not rectangular, or the principal point is not where it looks — the drawing is a crop, or was made with a shifted lens, and the middle of the sheet is not the middle of the picture.

The practical form is a second ruler measurement, and it costs nothing beyond the first. Having measured $a$ and $b$, compare them. If they are within a few per cent the symmetric rule holds and the table’s rows apply. If one is more than about twice the other, the picture is appreciably wider than the separation suggests and the honest reading distance is $\sqrt{ab}$ rather than half the separation — which is the same correction the two-point focal length applies when it declines to assume the principal point and computes it instead.

What to do about it #

For a draughtsman the instruction is short and it costs a beam compass.

Choose the field of view first, from the distance the picture will be looked at. Two page-widths of separation is a 53° picture and is about right for something held in the hand; three is a 37° picture and suits something on a wall. Then place the two points at that separation, off the sheet if that is where they fall, and construct the converging lines by one of the methods that do not need the meeting point.

For a reader, the instruction is shorter still, and it is the one this whole collection keeps arriving at: the picture states where to stand, and the statement is on the horizon. The distance from the centre of the picture to a vanishing point is a length that can be measured with a ruler, and half the separation of two perpendicular ones, scaled to the printed width, is the distance in centimetres.

Which means a reader with a ruler can find out, from any two-point drawing, exactly how close they were meant to be — and can find out that in most published examples the answer is closer than a person can focus.