Visual Plane
every picture states the point it is correct from
What survives

The diagonals find the middle

Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.
17 min read 12 figures Projected, not constructedWhat a projection destroysTaught, and never measured

Worth reading first: What a projection destroys · Where parallel lines meet.

A projection destroys length, angle, area, and the ratio in which a point divides a segment. That last one is the reason perspective is hard to draw by hand: the midpoint of a receding edge is not at the middle of the drawn edge, so every by-eye method for spacing a receding row of posts or a receding row of paving stones is a method for guessing at something the picture has thrown away.

Three of those methods are taught, all three are wrong, and the best misplaces a post by three and a half metres in a row that is supposed to be spaced at 1.4. There is a fourth, and it is not in the same category at all.

Halving a receding rectangle two waysThe diagonals cross at the image of the rectangle's centre, 8e-14 px from it — the construction is exact at every camera because it uses only which lines meet where, and that is what a projection keeps. Halving the drawn side with a ruler instead lands 22.7 px from the image of the side's midpoint.the diagonals against a ruler, at 3.2 mthe diagonals — exactthe ruler — 22.7 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000000 · construction 8e-14 px
Fig. 1 Two answers to “where is the middle of this receding rectangle”. The diagonals cross at the image of its centre, exactly. Halving the drawn side with a ruler lands 22.7 px from the image of the side’s midpoint, which is the drawn form of the same error the by-eye methods make.

Why the construction survives

Draw both diagonals of a rectangle and they cross at its centre. That is a fact about a rectangle in the world, and it is a fact of a particular kind: it says that three specific lines pass through one point. It contains no length, no angle, no ratio.

A projection maps lines to lines and preserves incidence — if three lines met before, their images meet, at the image of the meeting point. So the statement transfers to the picture unchanged, and the diagonals of the image cross at the image of the centre.

Not approximately. The construction lands on the projected true centre to arithmetic noise, at every camera the slider reaches and at every depth. The comparison in the figure is deliberate: the ruler’s answer is off by a distance the figure prints, and the construction’s is off by nothing, and the two are the same operation done in the two different geometries.

The harmonic set

There is a second reading of the same construction, and it is the one that generalises.

Take the near end of a receding side, the far end, the midpoint, and the side’s vanishing point. Those four collinear points have a cross-ratio, and it is −1-1 — exactly, at every camera. Four points in that relation are called a harmonic set, and the fourth is the harmonic conjugate of the third with respect to the first two.

So the midpoint’s harmonic conjugate is the vanishing point. Which is a strange-sounding sentence that says something practical: the midpoint of a receding segment is not a free choice on the drawn line, it is determined by the two ends and the vanishing point, by a construction with no measurement in it.

Four points on a line, before and after a projectionLength and the ratio of lengths do not survive the projection; the cross-ratio does, agreeing to 0e+0 relative.horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across
Fig. 2 The invariant the harmonic set is a special value of. Length and the ratio of lengths do not survive a projection; the cross-ratio does. A harmonic set is the case where the number is −1, which is why it can be constructed rather than measured.

The complete quadrangle

The general construction is older than perspective and needs only a straightedge. Given three collinear points AA, BB and CC, take any point off the line, draw the two rays through AA and BB, take any second point along the ray through CC, cross the four lines, and the remaining diagonal cuts the line at DD — the harmonic conjugate.

Two free choices are made in there and neither affects the answer. Taking four quite different auxiliary points gives the same DD to twelve significant figures, and the cross-ratio comes out at −1.000000000000-1.000000000000 in every case. That invariance is the theorem: a construction whose answer depended on the auxiliary point would be a drawing, and one whose answer does not is a projective fact.

The distance point at 620 px — a picture correct from 14 cmThe orthogonals go to the centric point and the diagonal goes to the distance point; the transversals are where they cross. The distance point's offset is the viewing distance, so moving it moves the reader, and the drawing gives no sign that anything has changed.centric pointdistance point, 202 px off the sheet →620 pxcorrect from 14 cm at 160 mm wide33° across
Fig. 3 The classical pavement, laid out with a distance point. It is exact and it needs the viewing distance to be chosen — which is what the diagonal construction does without.

Laying out a row with no measurement at all

Run the construction outward instead of inward and it becomes the method a draughtsman actually wants.

Halving a receding rectangle two waysThe diagonals cross at the image of the rectangle's centre, 8e-14 px from it — the construction is exact at every camera because it uses only which lines meet where, and that is what a projection keeps. Halving the drawn side with a ruler instead lands 22.7 px from the image of the side's midpoint. The same diagonal continued lays out 3 more bays of the same 2.1 m, with nothing measured.the diagonals against a ruler, at 4.2 mthe diagonals — exactthe ruler — 13.6 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000000 · construction 6e-14 px
Fig. 4 The same diagonal continued. Given one bay of a fence, the diagonal from a near corner through the midpoint of the far edge meets the extended side at the far corner of the next bay — and each new bay lands on the division the camera puts there, with nothing measured.

Given one bay of a fence — four corners, correctly drawn — every subsequent bay follows from a straightedge. Each new bay is checked against where the camera actually puts that division, and lands there.

There is a trap inside the construction that this figure fell into while it was being written, and it is worth recording because it is the essay’s own subject biting the essay. The construction needs the midpoint of the far edge, and the first version took it with a ruler — halved the drawn far edge. The far edge is a transversal seen obliquely, so the midpoint of its image is not the image of its midpoint, and the bays came out several pixels adrift by the third one. The midpoint has to be constructed too: the line from the rectangle’s centre to the receding direction’s vanishing point cuts both the near and the far edge in half.

The figure made the exact mistake it exists to point out, in its own machinery, and a numerical check caught it. That is the argument for checking a construction against the camera rather than looking at it, in one sentence.

Bands in a constant ratio, against the construction that replaced itBoth look like pavements. Asked what depth each drawn band claims, the constant-ratio rule gives 0.0, 1.0, 1.9, 2.8, 3.6, 4.2, 4.8 braccia where it should give 0, 1, 2, 3, 4, 5, 6 — it loses 1.2 braccia by the sixth band. The correct band ratios are not constant: they run 0.824 to 0.870, which is near enough to be mistaken for one.the construction — equal bracciaeach band 78% of the one beforeband 11.00 bracciaband 21.94 bracciaband 32.81 bracciaband 43.58 bracciaband 54.25 bracciaband 64.82 bracciawhat each band of the constant-ratio pavement claimsthe sixth band is 1.18 braccia shortboth drawings look like a floor
Fig. 5 One of the taught shortcuts, measured. A constant ratio between successive drawn depths is what the eye reaches for and it is not what a projection produces, so the row it lays out is a row of something else.

Against the measuring point

The measuring point is the classical construction for laying out correct depths, and it is exact too. It is worth saying what the difference is.

The measuring-point method needs the horizon, the vanishing point of the receding direction, and a measuring point placed at a specific distance along the horizon — which is to say it needs the station point, because that distance is the viewing distance. It gives depths in stated units: this post is at three metres, that one at four.

The diagonal construction needs none of that. It needs one bay already drawn, and it produces equal divisions — as many as are wanted, in either direction — without ever knowing where the eye is or what the units are. It cannot say that a bay is 1.4 m; it can only say that the next one matches this one.

That is a clean division of labour and it maps onto a distinction the stratification rung makes precise: equal divisions are an affine fact, recoverable from the vanishing line alone, and a length in metres is a metric fact needing more. The diagonal construction is the affine one and needs less. The measuring point is the metric one and needs the station point.

The measuring point, checked against the depths the camera produces5 equal depths of 1.40 m, laid out by the construction, land on the projected positions to 6e-14 px.24VPcorrect from 26 cm, at 160 mm wide34° across
Fig. 6 The classical construction, for comparison. It gives depths in metres and it needs the viewing distance to do it — the measuring point sits on the horizon at exactly the station point’s distance from the vanishing point.
The same cube turned 24° — 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 distance2564 px · 8129 px · 531 px
Fig. 7 The construction the diagonals are laid on. One, two and three point are one construction with the cube turned, and the diagonal method works in all three because it never asks how many vanishing points there are.

What the by-eye methods were reaching for

Look again at what the three taught methods try to do. Each takes a drawn interval and divides it by a rule that would be right if the picture were a scale drawing — halve it, take a constant ratio, interpolate toward the horizon. Each is an attempt to recover a length ratio, and length ratio is precisely what a projection destroys.

Three by-eye methods for spacing a receding row, against the truthThe posts are 1.4 m apart. The nearest by-eye method misplaces one by 11.72 m; the worst by 347.80 m.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
Fig. 8 One of the three, measured. The best of the taught methods misplaces a post by metres, because it is computing with a quantity the picture no longer carries.

The diagonal construction does not reach for a length ratio at all. It reaches for an incidence — these two lines cross here — and gets a length ratio as a consequence, because in the world the crossing point is the centre. That is the pattern, and it is the practical content of “the cross-ratio is the only surviving quantity”: a construction that only asks which lines meet where will survive; one that measures anything will not.

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. 9 The test that separates them. Four correctly projected divisions and four equally spaced ones can have the same cross-ratio, so the naive check passes both — the working version uses the vanishing point as its fourth point, and refuses what the weak one accepts.
Projective, affine, metric — what each stage buysThe photograph fixes the plane only up to a projectivity: the midpoint of a receding side lands 0.3970 of the way along. Supplying the plane's vanishing line buys the midpoint back exactly and nothing else. Supplying the image of one circle buys the last three numbers, at which point the right angle is 90.000° and two equal sides measure 1.000000. The cross-ratio is 1.333333 in all three, because it was never lost.projectiveaffinemetricmidpointtwo equal sidesa right anglecross-ratioprojective———1.333333333affine0.500000——1.333333333metric0.5000001.00000090.000°1.333333333— means the stage does not determine it at allcross-ratio 1.333333 throughout
Fig. 10 Where the construction sits in what a picture determines. Equal divisions are an affine fact and the vanishing line is what buys them; the diagonal method finds them without ever naming the vanishing line, because the fourth corner of the rectangle carries it.

What the construction is, stated projectively

The essay has been describing a drawing procedure. Stated as geometry, it is one sentence about incidences, and the sentence is what makes it exact.

A rectangle’s two diagonals meet at its centre. That is a claim that three lines — diagonal, diagonal, and any line through the centre — are concurrent, and concurrency is preserved by projection. So the image of the centre is at the crossing of the image diagonals, and no step of the argument involves a length, an angle or a ratio.

Compare what the taught methods claim. Halve the drawn interval claims that the image of the midpoint is at the midpoint of the image, which is a claim about a length ratio. Use a constant ratio between successive depths claims that the image of a geometric progression is a geometric progression, which is a claim about ratios of ratios. Both are false for the same reason: a projection destroys ratios along a line and preserves only the cross-ratio.

The by-eye methods are not sloppy versions of the diagonal method. They are attempts to compute with a quantity that is no longer there, and the diagonal method is the one procedure in the set that never asks for it.

What the harmonic set is, in the picture

The cross-ratio of −1-1 has a reading that makes the construction’s exactness feel less like a coincidence, and it is worth having because it explains why the vanishing point is the fourth point rather than some other.

Take a receding side of the rectangle and the four points on it: the near end, the far end, the midpoint, and the vanishing point. In the world, three of those are ordinary points and the fourth is at infinity — and the statement “the midpoint is the midpoint” is exactly the statement that the four are harmonic, because a midpoint’s harmonic conjugate with respect to the two ends is always the point at infinity of the line.

So the harmonic relation is not something the projection introduces. It is what being the midpoint means, once the point at infinity is admitted as an ordinary point. And a projection preserves the cross-ratio, so the relation survives into the picture with the vanishing point standing in for infinity.

That is the whole of the argument, and it is worth noticing that no step of it uses the fact that the figure is a rectangle. Any segment’s midpoint is the harmonic conjugate of its direction’s vanishing point; the rectangle is a convenience for constructing the relation with a straightedge, because its diagonals are already drawn.

Which is why the same construction transfers to shapes with no rectangle in them at all — to a chord of an ellipse, to a face of a box seen obliquely, to the two edges of a road. The relation is between four points on a line, and finding them is a matter of what happens to be drawn nearby.

The construction’s cousins in the drawing office

Three related constructions use the same fact and are worth naming, because a draughtsman who knows one usually knows all three without knowing they are one.

Halving. The diagonals of a bay find its centre, so the line from the centre to the receding vanishing point halves both the near and far edges.

Doubling. The diagonal from a near corner through the midpoint of the far edge meets the extended side at the next bay’s far corner — the essay’s repeat construction, run forwards.

And dividing into any number of parts. Divide the near edge, which is a transversal and can be measured with a ruler when it is parallel to the picture plane, then run each division to the receding vanishing point. That is the only one of the three that needs a measurement, and it needs it on the one edge where measuring is legitimate.

The three together lay out any regular grid in perspective with a straightedge and one ruler measurement on a frontal edge. That is what the classical perspective texts are describing when they talk about squaring the ground, and the reason it works is the reason this rung exists.

What a projective construction is worth

There is a general lesson in the comparison and it is the one this site keeps arriving at from different directions.

A construction that uses only incidence — which lines meet where — survives being photographed, because a projection preserves incidence and destroys everything else. A construction that measures anything does not survive, and the size of its error depends on how far the picture is from being a scale drawing.

So the question to ask of any drawing procedure is not whether it is accurate but which of the two kinds it is. The diagonal method is the first kind and is exact at every camera and every depth. Halving the drawn edge is the second kind, and at a long focal length from a distance it is nearly right, which is exactly what makes it survive: a rule that is close enough in the flat case gets taught, and the flat case is where perspective drawing is least needed.

The one thing it cannot do

The construction is exact and it is not free. It needs a correct starting rectangle, and it has no way of producing one — feed it a quadrilateral that is not the image of a rectangle and it will happily halve it, producing exact divisions of a shape that does not depict what was intended.

That is the cube that is a box as a hazard rather than a finding. The two-point construction every book teaches leaves the depth of the box undetermined, so a draughtsman who places the far edges by eye is drawing a box of some depth rather than a cube — and the diagonal method will then divide that box perfectly. A construction that is exact given its input can amplify a wrong input without ever complaining.

The taught two-point cube, with the two far edges placed 8 points apartThe corner angles are 90° because the method forces them. The side ratio is 0.719, so this picture depicts a box whose depth is 1.39× shallower than its width.horizoncorner angles90.000° — forced by the methoddepicted side ratio0.7195lens this drawing implies60° acrossdrawn exactly as the method prescribesthe free step is where the far edges go
Fig. 11 Where the wrong input comes from. Place the two far edges eight points apart instead of symmetrically — invisible on the page — and the drawing depicts a box 1.4 times shallower than it is wide. Everything constructed on it afterwards is exact about that box.

Which is the honest shape of every exact construction on this site. The projective ones cannot go wrong on their own; they can only propagate. What they buy is that the error is entirely in the input, where it can be found, rather than distributed through the drawing where it cannot.

Halving a receding rectangle two waysThe diagonals cross at the image of the rectangle's centre, 1e-13 px from it — the construction is exact at every camera because it uses only which lines meet where, and that is what a projection keeps. Halving the drawn side with a ruler instead lands 10.1 px from the image of the side's midpoint.the diagonals against a ruler, at 5.4 mthe diagonals — exactthe ruler — 10.1 px outcorrect from 30 cm, at 160 mm wideharmonic set -1.000000000 · construction 1e-13 px
Fig. 12 The construction from further away, at a narrower angle. The ruler’s miss shrinks as the picture flattens — at a long focal length from a distance, halving the drawn edge is nearly right — and the construction is exactly right at both ends, which is what makes it a rule rather than an approximation with a range.

Where this goes next

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. 246VPcorrect from 26 cm, at 160 mm wide34° across The measuring point, and the step the method leaves out Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose. horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across What a projection destroys A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

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.

Complete quadrangleCross ratioDepth divisionHarmonic conjugateMeasuring pointMidpointProjective invariantProjective stratificationTransversalVanishing point