The quadrilateral that finds the middle

The diagonals find the middle built the harmonic conjugate the way it is always built: four points off the range, joined up into a complete quadrangle, and the fourth point read off its remaining diagonal. Nothing is measured anywhere in it — only joins and meets — so it runs on a photograph with the camera unknown, which is what makes it the workhorse of this collection’s straightedge constructions.

Its dual is a construction from four lines. This essay runs it, and the interest is not that it works.

The construction #

Four lines in general position are a complete quadrilateral: six vertices where they cross in pairs, and three diagonals joining opposite vertices. The three diagonals form a triangle, and on each of them the two vertices it carries and the two points where the other two diagonals cut it form a harmonic range.

So: take the range A B as one diagonal. Draw two lines through A and one through B; the fourth is then forced, being whichever line through B makes the second diagonal pass through C. The third diagonal cuts A B at D, and (A B; C D) = −1.

Three lines free, one forced, four straightedge steps after that. The quadrangle construction is two points free, four joins and three meets. Same instrument, same length, different marks.

What is measured #

Three numbers, and the third is what makes the first two evidence.

The cross-ratio comes out −1.00000000000000. Two quadrilaterals built with different free lines land on the same fourth point to 1.2 × 10⁻¹⁵ of the range’s own length — which is the claim that the construction depends on the theorem and not on the choices. And moving C by a tenth of the range moves D by 2.18 range-lengths, which is what says the agreement above is a measurement rather than a routine that returns the same point whatever it is given.

That third number is the one this collection has learned to insist on. The polar with a straightedge records the same discipline for the same reason: two lines built from four points on a conic will usually be close to each other however they are chosen, so agreement is worthless until the construction has been shown to move.

The refusal #

Hand the construction the midpoint of A B and it refuses, because the third diagonal comes out parallel to the range and there is nothing to mark.

That is the theorem rather than a failure. The harmonic conjugate of a midpoint is the point at infinity — which is the fact that makes the whole family of constructions work, since it is how a straightedge alone can find a midpoint in a perspective drawing without measuring anything: join to the vanishing point, which is the harmonic conjugate’s own partner.

The quadrangle construction refuses at exactly the same input for exactly the same reason. Two dual constructions that fail on the same case is a stronger check than two that succeed on the same case, and it is cheap to run.

The refusal is a pole, and the pole has a neighbourhood #

The refusal at the midpoint is the end of a divergence rather than an isolated case, and writing the map down shows how large the neighbourhood is.

Put $A$ at 0 and $B$ at 1 and let $C$ sit at $t$. The harmonic conjugate is

$$D \;=\; \frac{t}{2t-1},$$

which is 0 at 0, 1 at 1, and infinite at $t = \tfrac12$ — the refusal, derived. Differentiating,

$$\frac{\mathrm dD}{\mathrm dt} \;=\; \frac{-1}{(2t-1)^{2}},$$

so an error in locating $C$ is amplified by $1/(2t-1)^{2}$ on its way to $D$. That is 1 at the ends and unbounded at the middle: 25 at $t = 0.4$, and 100 at $t = 0.45$.

Which is unfortunate, because near the middle is where a draughtsman most often wants the construction. Halving a bay, finding the centre of a panel, dividing a receding row — all of them put $C$ near the midpoint, and all of them are the badly-conditioned end. The construction is exact everywhere and usable at the ends, which is the collection’s recurring shape arriving on its shortest construction.

So run it the other way #

The same expression says the remedy, because the map is its own inverse and the amplification is therefore reciprocal.

Taking $C$ at or near infinity — that is, using a vanishing point as the given third point — puts $t$ large and the amplification at $1/(2t-1)^{2} \to 0$. The output is the midpoint, and it is found with an error smaller than the input’s rather than larger.

So the two directions of one construction have opposite conditioning:

Given a vanishing point, find a midpoint — amplification below one, and the further off the vanishing point the better. This is the direction the pavement and bay constructions use.

Given a near-midpoint, find its conjugate — amplification above one and unbounded. This is the direction to avoid.

Which turns a warning into a rule with no exceptions worth stating: always supply the point that is furthest along the line and construct the one that is nearest. A harmonic set has four points and the construction can be run to produce any of them from the other three; choose the run whose output is the crowded one and whose inputs are the spread-out ones, and the arithmetic works in the draughtsman’s favour instead of against them.

That also explains why the vanishing point is such a useful thing to have on a drawing, beyond its role in fixing directions. It is the one point of a range that is always available and always far away, so every harmonic construction that uses it is running in its well-conditioned direction — and a drawing without one is a drawing whose harmonic constructions must all be run the hard way.

Why the dual is worth having #

The theorem is the same. The marks are not, and on a real drawing the marks are the whole question.

A photograph of a colonnade gives long straight edges — the cornice, the base line, the top of the shafts — and very few securely locatable points. Four lines is what such a picture supplies. A photograph of a studded facade or a tiled floor gives the opposite: a scatter of marks and no long edges at all.

So the practical rule is short and it is not about geometry. Run whichever half of the theorem the drawing’s own evidence is made of. Duality guarantees that the other half exists at the same length and with the same instrument, so no drawing is ever the wrong shape for the construction.

The conditioning is not dual #

Duality carries the theorem across exactly and carries nothing about accuracy, which is the same finding five tangents name the same conic makes about a fit and is worth making again about a construction.

Both constructions have free choices, and both have choices that are bad. In the quadrangle version, a point placed nearly on the range gives a nearly degenerate quadrangle and a fourth point that moves a long way for a small slip; in the quadrilateral version, a line lying nearly along the range does the same. The bad cases are duals of each other, which is neat, and neither is predicted by the theorem.

What decides accuracy is how far the construction’s intermediate marks are from being degenerate, and that is a property of the choices a draughtsman makes rather than of which half of the theorem is being used. The slider on the figure above walks through those choices, and the fourth point does not move — but the intermediate vertices run right off the page at the extremes, and on a real drawing that is where the error would be.

The harmonic range is the same object in both #

A harmonic set of four is the smallest projectively meaningful configuration on a line — four points with cross-ratio −1 — and it is self-dual in the sense that four concurrent lines with pencil cross-ratio −1 are the dual object and behave identically.

That is why the construction can be run in a pencil rather than on a range and pulled back: the fourth harmonic ray of three concurrent lines meets the range in the fourth harmonic point of the three points they cut. Four lines have a cross-ratio is where this collection established that the range’s number and the pencil’s number are one number, and everything here follows from it.

Which means the two constructions are not merely both true; they are computing the same projective invariant of the same object, approached from the two directions the object can be described in.

What “forced” means, and why three lines are free #

The construction has four lines and only three of them are chosen. That asymmetry is the same one the quadrangle version has — two points chosen, and then everything determined — and it is worth spelling out because it is where a reader’s own attempt usually goes wrong.

The four lines of a general complete quadrilateral cut out a harmonic range on each of its three diagonals, but the range is whichever one they happen to cut out. To make the range the one wanted — with A, B and C in it — three of the four lines can be placed anywhere and the fourth has to be the one that drags the second diagonal onto C. Finding it is one join and one meet: join C to the vertex where the first and third lines cross, see where that line cuts the second, and join the result to B.

Everything after that has no choice in it at all, which is what makes the agreement between different choices a test of the theorem. If the fourth line were also free, two runs would land on two different points and there would be nothing to check.

The same construction on a photograph #

None of the steps measures anything, so all of them survive a projection, which is the property that makes this collection use straightedge constructions at all rather than arithmetic on coordinates.

Photograph a wall with three collinear marks on it, run the quadrilateral construction on the photograph, and the fourth point found in the picture is the image of the fourth point in the world. No focal length, no station point, no known length, no rectification. That is a strong statement and it is the same one the polar with a straightedge makes about a conic and the bay repeated by a straightedge about a colonnade.

The limit is worth stating with it: the construction finds a projectively defined point. If what is wanted is a midpoint in the world, the construction gives it only when the fourth harmonic point is the vanishing point — which is the case where the whole apparatus collapses to a single join, and is why the vanishing point is worth so much.

Where the fourth point ends up #

Worth a paragraph because the answer is counterintuitive and the figure shows it plainly.

As C moves from A toward the midpoint, D runs from A outward and off to infinity. As C passes the midpoint and continues toward B, D reappears from the far side beyond B and comes back in. The pair (C, D) sweeps out the whole line and passes through infinity exactly once.

That is an involution on the line — the harmonic involution with respect to A and B — and it is the object perpendicular is a pairing and the map a row of posts is are both about. The construction here is one evaluation of it; the involution is the whole family.

What a draughtsman would actually do #

Neither, mostly. On paper the quickest way to a harmonic conjugate is to use the vanishing point when there is one, which is the degenerate case where D is at infinity and the whole construction collapses to a single join.

The constructions matter where there is no vanishing point available — a picture of a plane with no repeated structure, a rectified image, a drawing whose edges do not run to any mark on the page. Then the choice between quadrangle and quadrilateral is a real one, and it is decided by the marks.

The one case where the dual is clearly the better tool: a range whose two ends A and B are off the page, defined only as the crossings of drawn lines. The quadrangle construction needs A and B as marks to join to; the quadrilateral construction never marks them at all, because they only ever appear as intersections that the construction’s own lines pass through.

What it inherits from Desargues #

The closure is not an accident of the drawing, and the reason it is not is Desargues read the other way.

Every step of both constructions is a join or a meet, so the question of whether they close is a question about incidence theorems in the plane the drawing is in. Desargues is the theorem that makes those closures hold on any plane sitting inside a three-dimensional projective space, which is every plane a reader will ever draw on. Its self-duality is what guarantees that the dual construction closes whenever the original does, without needing to be checked separately.

So the measurement in this essay is, strictly, redundant: the theorem says it must come out. Running it anyway is the same discipline as computing a reflection two ways — an implementation is not a theorem, and the number that comes back is evidence about the implementation as well as about the geometry. On this site that check has caught an inverted camera basis, an empty tick array and a flipped face normal, none of which any theorem would have predicted.

What the two constructions cost in marks #

A small accounting, because on a drawing the cost is what decides.

The quadrangle version needs the three given points marked, two free points chosen, four joins, three meets and one final join — eleven straightedge operations, and it puts four new marks on the paper. The quadrilateral version needs the three given points, three free lines drawn, one join and one meet to force the fourth, then four meets and two joins — twelve operations, and it puts six new marks on the paper.

So they are the same length to within one step, and the difference is entirely in what is marked: the first wants points near the range and the second wants lines crossing it. On a crowded drawing that decides which is executable, and on a clean one it decides which leaves less ink.

The one asymmetry worth knowing is that the quadrilateral’s intermediate vertices can run a long way off the page at extreme choices of the free lines, where the quadrangle’s stay near the range. That is a conditioning fact rather than a step count, and it is what the slider on this essay’s figure is walking through: the answer does not move and the working does.

The short version #

The harmonic conjugate built from four lines is the dual of the harmonic conjugate built from four points, and it is the same length, the same instrument and the same answer: cross-ratio −1 exactly, agreement between different choices at 1.2 × 10⁻¹⁵ of the range, and a fourth point that moves 2.18 range-lengths when the third point moves a tenth of one.

It refuses the midpoint, because the midpoint’s conjugate is at infinity, and so does the original. What is worth carrying is not that the dual exists but that the choice between the two is decided by which marks a drawing has, and that every straightedge construction here comes with the same choice.

One last note on when to prefer which. The two constructions differ in what they need present rather than in what they return, so the choice is made by the picture and not by taste: the diagonals want a complete quadrilateral whose four corners are all on the page, and the quadrangle route wants only that the two pairs of lines cross somewhere reachable. On a photograph where the far corners of a bay run off the frame, the second is available and the first is not, and both are exact.