The diagonals find the middle

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.

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 $-1$ — 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.

The complete quadrangle #

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

Two free choices are made in there and neither affects the answer. Taking four quite different auxiliary points gives the same $D$ to twelve significant figures, and the cross-ratio comes out at $-1.000000000000$ 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.

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.

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 size of that trap is worth having, because it decides how visible the mistake is. For an edge whose two ends sit at depths $z_1$ and $z_2$, the image of the midpoint falls at the fraction $z_2/(z_1+z_2)$ along the drawn edge, so a ruler halving it is out by

$$\frac{z_2 - z_1}{2\,(z_1 + z_2)}$$

of the drawn length. An edge whose far end is twenty per cent further away gives 4.5 per cent — four or five pixels on a hundred-pixel edge, which is exactly the “several pixels” the third bay showed. And the expression has a ceiling of one half, so on a strongly receding edge the ruler’s midpoint can be wrong by almost the whole half-edge.

Two things about that number make the mistake hard to catch by looking.

It is small on the near edge and grows with depth. The first bay’s far edge is barely foreshortened, so the ruler halving is nearly right and the first constructed bay looks correct. The error appears at the second and third, by which point a draughtsman has stopped checking the step and started trusting it.

And it is a bias rather than a scatter. Every ruler-halving errs in the same direction — toward the near end — so the constructed bays march toward the horizon slightly too fast, and the drawing reads as a row of bays that are correct at the front and crowd too early. That is precisely the appearance dividing depth by eye produces, which is the point of this whole row: the trap turns a projective construction into a by-eye one at a single step, and the result is indistinguishable from never having constructed anything.

The general rule the trap illustrates is the collection’s oldest one. A projection destroys a ratio along a line, so every metric step smuggled into a projective construction is a mistake of this kind — halving with a ruler, marking seven tenths along a diagonal, spacing transversals evenly. The construction survives the projection only if every one of its steps is a join or a meet, and the moment one of them is a measurement the whole thing inherits the measurement’s error and none of the theorem’s protection.

Which is why the repair is not “measure more carefully”. It is to replace the measurement with a construction — here the centre-to-vanishing-point line, which is two joins and a meet and costs the same two strokes the ruler did. That construction is what the bay repeated by a straightedge uses throughout, and it is why that essay’s twelve bays land at $10^{-12}$ px while this figure’s first attempt drifted at the third.

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.

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.

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.

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.

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 $-1$ 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.

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.

That is worth one closing distinction, because it decides where to spend care. An approximate construction degrades gracefully and is wrong everywhere by a little; an exact one applied to a wrong input is right everywhere about the wrong thing, and offers no residual to notice it by. The first invites more careful drawing and the second invites more careful checking of the rectangle — and the second is the better trade, because a rectangle can be verified against something outside the drawing and a distributed error cannot.