The map a row of posts is

A row of posts recedes to a vanishing point. The drawn gaps between them get smaller and smaller and never reach zero, and the posts never reach the vanishing point.

Everybody knows that. It is usually explained by saying that the drawn size falls as one over the distance, which is true and is a statement about arithmetic. There is a statement about structure underneath it, and it is sharper.

Advancing a bay is a map #

Fix the world line the posts stand on, and fix its image. Every point of the world line has a drawn position, so there is a correspondence between the world coordinate and the drawn one — and that correspondence is a projectivity, as a line is a space of its own says, because it is a perspectivity from the eye.

Now compose. Walking one metre along the world line is a map of the world coordinate: $t \mapsto t+1$. Conjugating it by the world-to-picture correspondence turns it into a map of the drawn line to itself: given a post’s drawn position, produce the next post’s drawn position.

That map is a single object with three numbers in it, and it is applied over and over. The drawn row is one map iterated, not nine positions computed.

Which map, and what it holds still #

A self-map of a line has fixed points, found from a quadratic, and there are exactly three cases. Normalise the matrix to determinant one and the whole classification is in the trace:

• trace squared above four — hyperbolic: two real fixed points;
• trace squared exactly four — parabolic: one, counted twice;
• trace squared below four — elliptic: none that are real, the pair being complex conjugates.

The bay-advance map is parabolic. Its trace squared comes out four to twelve decimal places, and it is parabolic at every camera, because conjugation cannot change a classification and $t \mapsto t+1$ is parabolic by inspection.

Its one fixed point is the vanishing point. Computed from the map’s own quadratic and compared with where the camera puts the vanishing point, the two agree to six parts in ten million million of a pixel.

That is the sharper statement. The posts crowd toward the vanishing point and never arrive because a parabolic map has nowhere else to send anything. There is one point it holds still, everything else moves toward it, and nothing ever gets there because arriving would mean two points landing on one and a projectivity is a bijection.

And the other case, which is the useful one #

Advancing by a fixed distance is parabolic. Scaling by a fixed factor is not.

Take a world line with an origin and mark the points at $1, 2, 4, 8$ metres — a geometric row rather than an arithmetic one. Doubling the world coordinate is $t \mapsto 2t$, which has two fixed points, at nought and at infinity. Conjugated into the picture, it becomes a hyperbolic map with two fixed points: the vanishing point, and the drawn image of the origin the scaling is about.

A hyperbolic map has a number attached to it that a parabolic one does not: the multiplier, which is the factor by which it moves points in the coordinate that puts its two fixed points at nought and infinity.

The multiplier is a cross-ratio — of any point, its image, and the two fixed points — and that is the whole reason it matters here. A cross-ratio can be measured on a drawing with a straightedge and no ruler. So the world factor is readable off the picture, exactly, with no camera, no focal length and no scene measurement.

The normal form, and what it says about the gaps #

A parabolic map has a normal form and it is the one this collection already uses everywhere else.

Put the fixed point at $p$ and change coordinate to $u = 1/(x-p)$ — the reciprocal of the drawn distance to the vanishing point. In that coordinate the map is $u \mapsto u + c$: a translation, because a parabolic map is a translation in the one chart where its fixed point is at infinity, and conjugation carried $t \mapsto t+1$ into it unchanged.

So $u_n = u_0 + nc$, and the row’s drawn positions are

$$x_n = p + \frac{1}{u_0 + nc}.$$

That is depth read as a reciprocal, arrived at from the classification rather than from the projection formula. And it gives a test that needs no camera and no ruler beyond a straightedge: measure the drawn distance from each post to the vanishing point, take reciprocals, and the results must fall on a straight line. A row that fails that test is not a row of equal bays, whatever it looks like.

The gaps follow immediately:

$$x_n - x_{n+1} = \frac{c}{(u_0 + nc)\,(u_0 + (n+1)c)}.$$

The drawn gaps therefore fall as the square of the reciprocal, not as the reciprocal — which is worth saying because “the drawn size falls as one over the distance” is the sentence usually offered, and the gaps are differences of that, so they fall faster than the thing they are differences of. And because the sum telescopes, the whole infinite row occupies exactly $1/u_0$ of drawn length: an unbounded row of posts fits inside the finite mark between the first post and the vanishing point, with room left over for none of them to arrive.

Why crowding is slow, and when it is not #

The multiplier of a map at a fixed point — the derivative there — is $1$ for a parabolic map, always, and that number is the reason the crowding looks gentle. Approach at multiplier one is arithmetic in $u$, so the residual distance falls like $1/n$: the ninth post is about a ninth of the way in that the first was, not a ninth of the previous gap.

The contrast is the hyperbolic case on the same drawing. Doubling a distance, rather than pacing one out, is the map whose trace squared came to four and a half above, and a hyperbolic map’s multiplier at its attracting fixed point is strictly inside the unit circle — so the approach is geometric, and each doubling closes a fixed fraction of what remains. A row of bays and a row of doublings recede to the same vanishing point at completely different rates, and the classification says so before any arithmetic is done.

One more consequence, and it is the one the drawing office uses. The whole row is fixed by $u_0$ and $c$, and only their ratio matters once the vanishing point is known, so two drawn gaps determine every remaining gap. That is exactly what a measuring point does mechanically: it supplies the second condition, and the row is then a consequence rather than a series of judgements. It is also why the pair whose depth is zero has to be chosen and cannot be discovered — $u_0$ is a choice of origin, and the classification is silent about it. What the classification does fix is everything downstream of that choice: pick the origin wrongly and the row is still a row of equal bays, drawn from a different standpoint, because a translation composed with a translation is a translation and the trace never moved.

Why it explains an earlier result #

Seven is not a power of two measured what a straightedge can divide a receding line into, and found that repeated bisection by diagonals reaches the dyadic fractions and nothing else — never a third, however many times it is spent.

Read as maps, that result is a statement about orbits. Bisecting is one map applied repeatedly, and the orbit of a point under a single map is a sequence — it contains what it contains. Reaching an arbitrary fraction needs an operation that is not the same operation iterated, which is what the ladder construction supplies.

The exactness of the bay repeated by a straightedge reads the same way. Iterating a projectivity accumulates no error because the projectivity is the same object each time; iterating a rule of thumb accumulates error because each application is a fresh approximation.

The conjugation is the whole argument #

One step above deserves to be stated on its own, because it is what makes every claim here independent of the camera.

The map of the drawn line is built as $M \circ T \circ M^{-1}$, where $T$ is the operation in the world and $M$ is the world-to-picture correspondence. That is a conjugation, and conjugation does not change a classification: the trace of a conjugated matrix is the trace of the original, so a parabolic $T$ gives a parabolic result whatever $M$ is.

So “advancing a bay is parabolic” is not a measurement made at one camera and hoped for at others. It is a fact about $t \mapsto t+1$, transported. Move the eye, change the focal length, tilt the picture plane, put the row at any angle across the room — the drawn positions move and the classification does not.

What the camera does decide is where the fixed point lands, and that is exactly the vanishing point — the one quantity a camera contributes to the whole arrangement. The structure comes from the world operation and the position comes from the camera, and separating those two is the useful part of writing it as a conjugation at all.

The third case, which has no visible fixed point #

The elliptic case is the one a drawing office never notices and the one this site’s own machinery rests on.

A map of a line with trace zero is its own inverse — an involution — and its two fixed points are a complex conjugate pair. Nothing on the line stands still.

“No direction on a horizon is its own perpendicular” is an obvious sentence about geometry. “This map has no real fixed point” is the same sentence in the vocabulary of the line, and the two imaginary numbers it does have turn out to be the focal length and the centre of the picture — which is the next rung and is not obvious at all.

What a projection does to a sequence #

There is a small result here worth having on its own, because it is the answer to a question a reader of any perspective drawing eventually asks: can the world spacing be read back out of the drawn spacings?

For a row laid out at equal world intervals, the answer is yes and it needs three marks plus the vanishing point. Three drawn positions and the vanishing point determine the map — three pairs, since consecutive positions give two pairs and the vanishing point gives the third as a fixed point — and once the map is in hand every further post’s position is a prediction.

Which is where the trap dividing depth by eye records comes from, stated structurally. Four consecutive members of a sequence are not four independent points: three of them can be carried anywhere, so a number computed from all four is a number about the fourth alone. The vanishing point is the point that was not carried along, and putting it in is what turns a tautology into a test.

What the classification buys #

Three things, and each replaces a piece of reasoning that was being done by hand.

It settles a behaviour without computing it. A row of things laid out at equal world intervals crowds toward one point and never reaches it, in every picture, from every camera, and the reason is a trace rather than a limit.

It says which drawings support which measurement. An arithmetic row’s map has no multiplier, so there is no number to read off it; a geometric row’s does. A photograph of equally spaced fence posts and a photograph of a decaying sequence are not the same measurement problem, and the classification says so before anything is measured.

And it puts a name on the degenerate case. A drawn row whose map comes out with trace squared very close to four but not exactly is a row that was not laid out at equal intervals, and how far from four it is, is a measurement of how unequal.

The horizon is the same object, used differently #

One more place the map appears, and it is the one that connects this rung to the construction field’s own work.

A vanishing point is a point of the horizon. Turn a direction in the ground plane and its vanishing point moves along the horizon, and that motion is a map too: turning by a fixed angle is a projectivity of the horizon, applied once per turn.

It is not a rotation of the horizon and it does not move points at a constant rate. Turning through equal angles moves the vanishing point in unequal steps, running out to infinity as the direction comes parallel to the picture plane and returning from the other end. That is the pole again — the same reciprocal that throws a point to infinity in every other case here.

The map turning by a right angle performs is the special one, because turning twice by a right angle is turning by a straight angle, which puts a direction back where it started. Applied twice it is the identity — so it is an involution, so it is elliptic, and so it holds nothing on the horizon still. Everything the next rung does follows from that one sentence.

What this does not say #

It says nothing about the posts being upright, or equal, or visible. Everything above is about one line and the marks on it; the posts are drawn because a bare line with marks is hard to look at.

It says nothing about noise. Fitting the map from three drawn marks and reading its trace is exact here and is conditioned in practice, and how badly depends on where the three marks sit relative to the vanishing point — which the pole in the denominator already warns about.

And it does not make the classification a test on its own. A drawn row that fails to be parabolic was not laid out at equal intervals; a drawn row that is parabolic was laid out at equal intervals or was drawn by somebody following a rule that happens to produce a parabolic map, and there are such rules. Which is the same warning dividing depth by eye issues about the cross-ratio, arrived at from the other end.

It says nothing about what a viewer infers from a receding row. That the crowding reads as depth is a fact about seeing, and this site has no standing on it; what is computed here is only that the crowding is a map with a fixed point, and that the fixed point is where the row’s direction points.

And it does not claim the three cases exhaust what a drawing can contain. They exhaust the maps of one line to itself, which is a smaller statement. A drawing contains maps between different lines, maps of the plane, and constructions that are not maps at all — and the classification says nothing about any of them.

The transferable form #

A repeated construction is one map applied many times, and the map’s fixed points decide what the repetition can and cannot reach — before anything is computed and whatever the camera was.

Every claim above followed from a trace. The posts crowding to a point, the multiplier being readable with a straightedge, the bisection missing every third, the perpendicular pairing having nothing real to hold still: four different-sounding facts, one classification, and the classification is a quadratic’s discriminant.

The habit worth taking is to ask, of any drawing procedure that is applied over and over, which map is being iterated — because the answer is usually already written down, and it usually decides the question that was being approached by measurement.