Every projectivity is two perspectivities

A perspectivity is the simplest thing one eye can do to a line: put a centre somewhere, draw a line from it through each point of one line, and mark where each ray crosses another line. Nothing is measured, nothing is computed, and the whole operation is joins and meets.

A projectivity is anything that preserves the cross-ratio. It is a far larger class in appearance, and three kinds of map on a row of posts sorts its members by how a point moves under iteration.

The two classes are almost the same size, and that is this essay. Between two distinct lines, every projectivity whatsoever is a composition of exactly two perspectivities — never more, and one is not always enough. The construction needs no arithmetic, and its being available is the reason a straightedge suffices for constructions elsewhere in this collection that look as though they should need a ruler.

The lemma the construction turns on #

The construction has one idea in it, and everything else is bookkeeping. It is worth extracting because it is the step a reader is most likely to accept without seeing why it is true.

A projectivity between two distinct lines that holds their meeting point still is a perspectivity.

The proof is three lines and uses nothing but the fundamental theorem. Let the map take the line c to the line b, and let X be the point where they cross, with the map holding X. Pick two other pairs, P → P′ and Q → Q′, and let S be the point where PP′ crosses QQ′. Now compare the given map with the perspectivity centred at S. They agree at P, they agree at Q, and they agree at X — because a perspectivity holds the meeting point of its two lines, just as the given map does. Three agreements, and two projectivities agreeing at three points are equal. So the given map is that perspectivity.

Everything else in the construction exists to arrange the hypothesis. Putting the first centre S₁ on the join A₁B₁ is what makes A₁ land on B₁; and once the intermediate line c passes through B₁, the point B₁ is the meeting point of c and b, and the remaining map holds it. That is the whole trick: the first perspectivity is chosen not to do anything useful, but to leave behind a map of the one kind the lemma recognises.

Which is why sliding the intermediate line off B₁ destroys everything. It does not make the construction slightly worse — it removes the hypothesis, the remaining map no longer fixes the meeting point, and the joining lines stop having a common centre to pass through.

What is actually being tested, which is not the obvious thing #

The natural way to check this construction is to ask whether the composite sends the given points where the projectivity does. That test is worthless, and saying why is the most useful paragraph in this essay.

The composite is a projectivity, because a perspectivity is one and a composition of two is one. It agrees with the target map at the three given pairs by construction. And two projectivities of a line that agree at three points are equal — that is the fundamental theorem in one dimension, the same fact that makes three pairs enough to specify a map at all. So the fourth point closes no matter what the construction does, provided only that each step really was a perspectivity. Checking it proves the theorem it assumes.

What has something to prove is the second step’s being a perspectivity rather than merely a projectivity. A perspectivity has a single centre through which every joining line passes, so the honest measurement is concurrency: take points nobody used, join each to its image, and ask whether those lines meet at one point.

The 37.7 pixels is what makes the thirteen places a measurement. Without it, a figure showing lines that meet would be showing a property of the drawing rather than a property of the construction.

Both choices are free, and the composite does not notice #

The construction asks for two decisions that nothing determines: where the intermediate line leans, and where on the join A₁B₁ the first centre sits. A reader is entitled to suspect that a construction with two arbitrary inputs is producing an arbitrary answer.

It is not, and the sweep above is the evidence. The second centre S₂ is a real function of both choices and moves 663 pixels as they vary — this is not a case where the free parameters quietly cancel before anything happens. What is invariant is the composite map, which is the same projectivity at every setting, because it is pinned by the three pairs it was built from.

That is the ordinary situation with a good construction and it is worth naming as a shape: a free parameter that visibly moves the intermediate objects and provably does not move the result. The measuring-point construction elsewhere in this collection has the same shape, and so does the inaccessible vertex, whose whole virtue is that the vertex is not one of its inputs.

Counting the parameters says how many are needed, before any drawing #

The number two is not arbitrary and it can be predicted by counting, which is a better reason to believe it than watching one construction close.

A projectivity between two given lines is fixed by three pairs, so it has three degrees of freedom. A perspectivity between the same two lines is fixed by its centre, a point of the plane, so it has two. Two is less than three, so a single perspectivity cannot reach every projectivity — and the shortfall is one parameter, which is exactly the one condition the previous section identified: holding the meeting point still.

Compose two perspectivities and there are four parameters, two centres. The composites they produce form a three-parameter family, because that is all the projectivities there are. So one parameter of the four must be redundant, and the redundancy is not an accident of the construction — it is forced by the arithmetic before anybody draws anything.

That is what the sweep is showing. The second centre travels 663 pixels while the composite stands still, and the count says it had to: there is a one-parameter family of decompositions of every projectivity, and the free choice of where the intermediate line leans is a coordinate on it. A construction whose free parameter moved the answer would be contradicting the count rather than merely being wrong.

The same arithmetic says why three perspectivities are needed for a map of a line to itself. There is no second line, so one has to be borrowed and returned from, and the borrowing costs a step.

One perspectivity is not enough, and the deficit is computable #

The theorem says two. It is worth showing that one will not do, and the interesting part is that the failure is not a residual — it is a definite position, predictable before the straightedge is picked up.

A perspectivity between two lines has two degrees of freedom, its centre, so two pairs fix it and the third is a prediction. But it also cannot help fixing the point X where the two lines cross, since X is on both and its ray through the centre is the line joining it to itself. So a perspectivity holds X still, whatever its centre — and a projectivity in general does not.

So the obstruction has a name. A single perspectivity can realise exactly those projectivities that already hold the meeting point of the two lines, and the amount by which a given map fails to be one is the amount by which the cross-ratio with X differs between the two lines.

That also says where the exception lives, and the exception is real rather than a limiting case.

Between the two settings nothing changes but where the third pair sits. The construction, the lines and the method are identical, and the answer moves from 154.98 pixels to 0.02 — because the condition being tested is a property of the data rather than of the technique.

Why this makes a straightedge sufficient #

The reason to care is not the theorem. It is what the theorem licenses.

Every step of the construction above is a join or a meet. Nothing is measured, no length is transferred, no angle is used, and no arithmetic is done. A projection preserves joins and meets exactly — that is what a projection is, at the level where this collection works — so the whole construction survives being photographed.

This is the fact underneath a great deal of what this collection does with a straightedge. What a straightedge reaches on a receding line measures the reachable set and finds it to be exactly the rational points; the reason the straightedge gets anywhere at all is that it can execute a perspectivity, and the reason it gets everywhere the cross-ratio allows is that two perspectivities make any projectivity. The two essays are the two halves of one statement: this one says what the instrument can do, and that one says how far doing it repeatedly gets.

It is also why the bays that are not equal can be laid out at all. An unequal spacing is a projectivity that the diagonal cannot reach, and the measuring-point construction reaches it because it is a perspectivity from a chosen centre followed by another.

The dual half, which costs nothing #

Everything above was about points on lines. The dual statement is about lines through points, and it is true for free.

Exchange point and line throughout and a perspectivity between two lines becomes a perspectivity between two pencils: instead of a centre through which joining lines pass, an axis on which meeting points lie. The decomposition theorem dualises with it, and a construction carried out in one reading can be read off in the other without being redone. A point and a line are one object is where that exchange is measured.

What the two centres are, when the lines are a photograph #

It is worth saying what the centres correspond to when the two lines are things in a room rather than marks on a diagram, because the abstraction hides something concrete.

A perspectivity between two lines is exactly what a pinhole does to them. Put an eye at S, and the map carrying each point of one line to where its ray meets the other is a perspectivity with centre S — so a perspectivity is not merely like a projection, it is one, restricted to a line. Where parallel lines meet is the same operation with one of the lines at infinity.

That gives the theorem a reading in the room. Any projectivity between two lines, however it arose — a shadow, a reflection, a rectification, a change of camera — can be realised by two successive photographs, taken from two computable stations, with an intermediate line to catch the first. The map does not remember which of the many possible pairs produced it, and the sweep is the family of pairs it does not remember.

This is the same shape of claim as a shadow being a second projection and a mirror being a second camera: a construction that looks like its own kind of thing turns out to be the site’s one operation with the centre moved. Here it is that operation applied twice.

Where the two lines coincide, and the count goes to three #

One case is deliberately excluded above and it is the one this collection meets most often.

The theorem says two perspectivities between two distinct lines. If the projectivity is a map of a single line to itself — which is what three kinds of map on a row of posts classifies, and what a repeated bay is — then there is no second line to project onto, and the decomposition needs an auxiliary one. Going out to a helper line and back is three perspectivities in general, not two.

That is not a defect in the theorem; it is the reason the two situations are worth keeping apart. A map between two lines has no fixed points to count, because a point of one and a point of the other are not comparable. A map of a line to itself has the classification, and the price of having it is one more perspectivity in the decomposition.

What the decomposition is not #

Two readings of the theorem are natural and both are wrong, and they are wrong in opposite directions.

The first is that the two perspectivities are somehow in the projectivity, waiting to be found — that a map arrives carrying a preferred pair of centres. It does not. The sweep is the disproof: 663 pixels of second centre, every position equally correct, and no principle in the geometry selecting one. A projectivity is a map, and its decompositions are things a reader constructs rather than things it contains.

The second is the reverse — that because the decomposition is not unique it is not telling anybody anything. That is also wrong, and the useful way to see why is to ask what the theorem forbids. It says no projectivity between two lines needs three. There is no map so awkward that a third step is required, no residual class of transformations lying beyond the reach of two applications of one eye. A closure statement of that kind is informative precisely because it is a bound, and bounds do not become vacuous by having many witnesses.

The distinction matters for how the result is used elsewhere. When a straightedge reaches exactly the rational points of a drawn line, the argument runs through what a straightedge can execute, and it needs the bound rather than a particular decomposition. Any two perspectivities will do; that there are infinitely many is what makes the argument robust rather than delicate.

The construction against the algebra, which is the site’s usual pairing #

There are two ways to get the image of a point under a projectivity, and this collection prefers to have both wherever it can.

The algebraic route takes the three given pairs, solves for the two-by-two matrix up to scale, and applies it. It involves a determinant, a division, and a coordinate system that somebody has to choose. The constructive route is the one drawn above: two centres, a straightedge, and no coordinate system at all.

They agree to 1.2 × 10⁻¹³ pixels at the first figure’s setting and 2.8 × 10⁻¹⁴ at the second, and the agreement is worth something because the two share no arithmetic. The matrix route never draws a line; the straightedge route never forms a product. A coding error in one is very unlikely to be reproduced by the other, which is the whole reason for computing a thing twice.

It also settles which route is the primary one for this collection’s purposes. The matrix needs coordinates, and coordinates are a choice a reader makes about a picture rather than something the picture supplies. The construction needs nothing but the marks, which is why it survives being photographed and why the tilted board closes as tightly as the flat diagram does.

What this does not settle #

The construction is exact and it is not stable in the sense a numerical analyst would want. Every step is a meet of two drawn lines, and two lines that cross at a shallow angle locate their meeting point badly. Choosing the intermediate line’s lean badly does not move the composite — the sweep proves that — but it does move how precisely a hand executing the construction can find S₂. Nothing here measures that, and the honest statement is that the theorem is about what is constructible rather than about what is drawable with a pencil of finite width.

And the decomposition is not unique, which the sweep makes vivid: 663 pixels of second centre, all of them correct. There is no canonical pair of perspectivities to point at, only a two-parameter family of them, and any statement of the form “the two perspectivities of this map” is a statement about a choice somebody made.

One eye, twice #

The object this essay is about is a single operation — put a centre down, draw rays, mark where they cross another line — and the finding is that doing it twice is enough for everything the cross-ratio permits.

That is a strong closure property and it is worth setting beside the alternative. If two perspectivities were not enough, a straightedge would be a weak instrument and the constructions in the construction field would need a ruler. Because they are, an entire tradition of drawing gets its correctness from joins and meets alone — and gets to keep it under photography, since a photograph preserves exactly those.