Visual Plane
every picture states the point it is correct from
Drawn confidently

Both vanishing points on the paper

Putting the two vanishing points on the sheet is presented as a composition rule. It is a statement about the reader: with the two points one page-width apart the picture is a 90° view, correct from 80 mm, and a reader holding it at arm's length is shown a room five times as deep as the one drawn. The layout that is honest at arm's length puts both points four and a half pages off the sheet.
17 min read 13 figures Taught, and never measuredThe viewer is in the geometryOne picture, one length

Worth reading first: The point you have to stand at · One, two and three point are one construction · The sixty-degree cone of vision.

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.

What the composition rule commits the reader toWith the two points one page-width apart the picture is correct from 80 mm — a 90° field of view — and a reader holding it at 400 mm sees a room 5.0 times as deep as the one drawn. Nothing about the drawing changes; the number is the layout's.0.6× the page48 mm×8.3 at 4000.8× the page64 mm×6.3 at 4001.0× the page80 mm×5.0 at 4001.4× the page112 mm×3.6 at 4002.0× the page160 mm×2.5 at 4003.0× the page240 mm×1.7 at 4004.5× the page360 mm×1.1 at 4007.0× the page560 mm×0.7 at 400distance the picture is correct from, shown 160 mm widea rule about the paperwhich is a rule about the reader
Fig. 1 Vanishing-point separation, expressed in page widths, against the distance the resulting picture is correct from when it is printed 160 mm wide — with the depth exaggeration a reader holding it at 400 mm receives. Both points on the page is 80 mm and a room five times too deep.

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.

Where the reader has to be for a 90° picture to be correctShown 160 mm wide, this picture is a correct projection only from 8 cm away. Drawn to scale.the picture, 160 mm wide8 cm90°the eyefocal length 345 px8 cm at 160 mm wide
Fig. 2 The reader’s own position, drawn to scale, for the field of view that layout produces. The picture width and the correct viewing distance are the same kind of quantity and are drawn at one scale, so the triangle is the one the reader’s eye would have to make.
Where the reader has to be for a 53° picture to be correctShown 160 mm wide, this picture is a correct projection only from 16 cm away. Drawn to scale.the picture, 160 mm wide16 cm53°the eyefocal length 692 px16 cm at 160 mm wide
Fig. 3 The reader’s position for a layout two page-widths wide, drawn to the same scale as the one above. This is the arrangement a picture held in the hand wants, and it puts both vanishing points off the sheet.

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 same picture, read from 40 cm instead of 19 cmNot one mark has moved: the reconstruction re-projects onto the drawing to 6e-14 px. What has changed is the solid the drawing depicts — a cube at 18.8 cm, and 2.12× as deep as it is wide from 40 cm.the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 2.12, width × 1.00correct from 18.8 cm at 160 mm wideread from 40 cm — depth × 2.12
Fig. 4 The same picture read from the wrong distance, and the scene that reading depicts. The paper is untouched; the room behind it has been stretched along the line of sight, and by exactly the ratio of the two distances.
A 90° picture, 160 mm wide, read from various distancesThe picture is correct from 8.0 cm. Read from an ordinary reading distance of 40 cm it depicts a scene 5.00× deeper than the one it was made from, and no mark on the page has moved.051020406080100how far the reader's eye is from the page (cm)how much deeper the depicted scene becomescorrect at 8.0 cm40 cm → × 5.00at 160 mm wide× 5.00 at 40 cm
Fig. 5 The exaggeration against the reader’s distance. It is a straight line through the origin, which is what makes the rule so simple to state and so easy to overlook: twice the distance, twice the depth.
The same picture, read from 80 cm instead of 19 cmNot one mark has moved: the reconstruction re-projects onto the drawing to 6e-14 px. What has changed is the solid the drawing depicts — a cube at 18.8 cm, and 4.24× as deep as it is wide from 80 cm.the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 4.24, width × 1.00correct from 18.8 cm at 160 mm wideread from 80 cm — depth × 4.24
Fig. 6 The same substitution at twice the reading distance. Depth scales with the distance and nothing else changes, so the two figures differ by a factor and not by a kind.

The table

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

points apart field of view correct from depth a reader at 400 mm sees
0.6 page 118° 48 mm ×8.3
0.8 page 103° 64 mm ×6.3
1 page 90° 80 mm ×5.0
1.4 pages 71° 112 mm ×3.6
2 pages 53° 160 mm ×2.5
3 pages 37° 240 mm ×1.7
4.5 pages 25° 360 mm ×1.1
7 pages 16° 560 mm ×0.7

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.

The vanishing point runs to infinity and the measurement does not careAs the camera comes level the vertical vanishing point leaves the canvas, the page and eventually the plausible — 7.2 × 10⁹ px at a tilt of one part in eight million. The recovered height stays exact to 2e-16 relative the whole way. At exactly level the method has nothing to work with and refuses.0510-4-3-2-10how far the camera looks down, over eight metres (metres, log scale)where the vertical vanishing point falls (log₁₀ pixels)the vanishing pointthe error in the recovered heightthe error curve is offset by 17 decades to be visiblea flat line at machine precision
Fig. 7 The problem the advice exists to dodge, drawn honestly. A vanishing point outside the drawing is not an obstacle to the geometry; it is an obstacle to the pencil, and the repair is a different construction rather than a different scene.

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”.

Seven identical spheres across a 90° frameThe outer sphere images 27% wider than the central one. That is what a correct rectilinear projection does, and it vanishes if the picture is viewed from 8 cm.49 px62 px90° across27% wider at the edge
Fig. 8 What a 90° layout does to a sphere at the edge of the frame. The elongation is what the projection requires and it reads as an error from anywhere but the station point — which for this picture is 80 mm from the paper.
Edge stretch against field of viewThe stretch is 1/cos θ at the frame edge: 2% at 24°, 13% at 55°, 39% at 88°. None of it is a lens fault.020406020406080100horizontal field of view (degrees)how much wider a shape images at the frame edge (%)2%13%39%1/cos θ at the frame edgea property of the projection, not the glass
Fig. 9 And the cure, which is not to redraw anything. Read from the distance the picture is correct from, the marginal stretch is exactly zero. The complaint is about where the reader is standing.

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.

What the sixty-degree cone is a rule aboutA 60° picture is correct from 0.866 of its own width — 13.9 cm at 160 mm wide. A reader at an ordinary 40 cm is 2.89× too far back and sees a scene 2.89× too deep. The rule cannot fix that; it only makes the error smaller by making the pictures narrower.02.5057.5010255075100125field of view of the picture (degrees)how much deeper the scene looks, read from 40 cmno error60° → × 2.89160 mm wide, read from 40 cm60° is correct from 13.9 cm
Fig. 10 The rule restated as what it is. The 60° cone is a claim about how far the reader will be from the page, and it can be checked against how far readers actually are — which is the only kind of evidence it could ever have had.

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.

Level, tilted, shiftedTilting the camera up to fit the building in makes the verticals converge by 4.20°. Shifting the lens up instead moves the principal point 95 px off centre, frames the same view, and leaves the verticals parallel to 0e+0° — because the picture plane never tilts.level — the top is cut off0.00° of spreadtilted 12°4.20° of spreadshifted 95 px0.00° of spreada shift moves every point by exactly the shift95.0 px, and no direction at all
Fig. 11 The camera that produces an off-centre principal point on purpose. Shifting the lens instead of tilting the camera keeps the verticals parallel and moves the optical axis off the middle of the frame, which is the honest version of an asymmetric layout.

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.

How far back the picture is correct from, against how wide it isA 37° picture is correct from 24 cm and a 90° picture from 8 cm.020406020406080horizontal field of view (degrees)correct viewing distance for a 160 mm wide picture (cm)a normal photograph — 24 cmwide — 16 cmvery wide — 8 cmsame picture width throughoutthe only variable is the angle
Fig. 12 The one curve everything here is on. Field of view against the distance a picture is correct from, at a stated printed width. A vanishing-point layout is a point on this curve whether the draughtsman thought of it that way or not.
The same cube turned 45° — 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 distance1150 px · 8129 px · 1150 px
Fig. 13 And the reminder that the number of points was never the choice. One, two and three point perspective are the same construction with the box turned; what a draughtsman is choosing when they place two points on a page is not a style of perspective but a lens.

Where this goes next

the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide The point you have to stand at A perspective picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm. Nobody stands there, and that single fact explains most of what gets called distortion. the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 2.12, width × 1.00correct from 18.8 cm at 160 mm wideread from 40 cm — depth × 2.12 Standing in the wrong place A picture read from twice the distance it is correct from depicts a scene twice as deep — and not one mark on the paper moves. The error is invisible in the picture, which is why it survives everywhere. drawn: 1.309 : 1subtended at the station point: 1.000000000 : 1the faint circle is the mean radius, for comparisonthe drawn centre sits 3.42 px from the axis's own mark The sixty-degree cone of vision Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.

What links here

Computed from the collection, not written here: the essays that point at this one.

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.

Compositioncone of visionDepth scalingfield of viewFocal lengthPicture planePrincipal pointStation pointVanishing pointViewing distance