The bay repeated by a straightedge

Every drawing manual gives the same construction for a row of fence posts, a line of columns or a run of paving. Draw the first bay. Cross its diagonals to find its centre. Run a line from the centre to the vanishing point to reach the midpoint of the far edge. Then draw from a near corner through that midpoint, and where it meets the opposite receding line is the far corner of the next bay.

Repeat. No measuring, no arithmetic, and no reference to the eye’s distance.

Checked, not admired #

This site’s habit with a taught construction is to draw it beside the projection it claims to produce. Two of them have come out badly: dividing depth by eye misplaces a post by metres, and the cube drawn by eye is a box.

This one comes out exact. After eight bays the worst constructed corner is 8e-13 px from where the camera puts it, and after twelve it is 1e-12 px. The residual is growing in the way rounding grows, which is to say hardly, and it is fifteen orders of magnitude below anything a draughtsman could draw.

Why it is exact #

The construction is a homology of the picture, and once that is said the exactness is not surprising.

Take the map that carries each bay to the next. In the world it is a translation by one bay’s depth. In the picture it is the image of that translation, which is a projective map of the picture plane with two pieces of fixed structure:

The vanishing point of the receding direction is fixed. Translating along a direction does not change where that direction vanishes.

And the line at infinity’s image — the horizon — is fixed pointwise. A translation carries every horizontal direction to itself, so every point of the horizon stays put.

A projectivity with a line of fixed points and a centre off it is a homology, which is the object the census of this site’s plane maps found under a shadow, a floor anamorph and a mirror. This is a fourth instance, and it is the one with an axis on the horizon.

So the bay-to-bay map is a homology; the construction executes it with a straightedge; and iterating a homology gives a homology. There is nothing to accumulate.

The diagonals are the harmonic construction #

The step that finds the midpoint is worth naming, because it is a result the foundations field owns and this is its most familiar use.

The centre of a rectangle is where its diagonals cross, and the line from that centre to the vanishing point cuts the far edge at its midpoint. In projective terms, the midpoint of a segment is the harmonic conjugate of the point at infinity with respect to the segment’s two ends — and the harmonic conjugate is constructible with a straightedge alone.

Which is exactly why the manual’s construction needs no measurement. Finding a midpoint sounds metrical, and the midpoint of a segment in the picture is not what is being found: what is found is the image of the world’s midpoint, which is a projective construction on four points and does not know what a length is.

Where it stops #

The bays crowd toward the horizon, and the construction runs out with them.

Each bay’s far edge is nearer the horizon than the last, by a factor that tends to a constant — the bays form a geometric sequence in the picture, which is what a homology iterated does. In the figure the gap between the first pair of edges is 26 px and between the last pair is 3.6 px after twelve bays.

That is a statement about the picture’s resolution rather than about the method. Once two consecutive far edges are within a pixel, the diagonals cross at a point the drawing cannot resolve, and the construction is reporting rounding. Its arithmetic goes on being exact long after its drawing stops being usable — which is the same separation the light field measures when a lamp is walked toward infinity, and the same one a stereo pair’s depth has past its range.

Exact everywhere, drawable over a range, and the range is set by the picture and not by the geometry.

What it does not need, and what that says #

The list of things the construction does without is the reason it survived four centuries of drawing manuals.

No focal length. The eye’s distance never enters. The construction works on whatever picture it is given, and produces the correct continuation of that picture.

No measurement. No ruler, no dividers, no scale bar. A straightedge, and lines through points already on the paper.

No knowledge of the bay’s size. The first bay is whatever it is, and the rest follow.

And that combination says something about what perspective construction is. The repetition is projective — it belongs to the group of maps preserving incidence — and everything projective is constructible with a straightedge. The constructions that need more, like the measuring point, need it exactly because they are supplying metric information the picture did not have: a real length, transferred in from the world.

The half-bay, and other divisions #

The same machinery divides as well as repeats, and the two are the same operation run in opposite directions.

Halving. The diagonals give the midpoint, so a bay can be split into two, then four, then eight, exactly, by straightedge alone.

Thirds and other fractions. Available too, by a longer construction — the projective one that divides a segment in any rational ratio using a pencil of lines. Slower with a pencil, exact with the same justification.

And the reverse. Repeating backwards toward the viewer is the same homology inverted, which is another homology, so a row can be extended in front of the first bay as easily as behind it.

What is not available is dividing a bay into a number of equal parts by eye, which is the construction the wrong field measures. The difference is not care: one operation is the exact execution of a projective map and the other is a guess whose error is measured in metres.

What the ruler would have said #

The comparison that makes the construction’s exactness meaningful is with the obvious alternative, which is to measure.

Lay a ruler on the drawing, find the middle of the drawn far edge, and mark it. That point is the centre of the image, and the image of the world’s centre is somewhere else — 25.4 px away in the figure above, on an edge a few hundred pixels long.

The gap is the perspective divide doing what it does: a ratio along a line is destroyed by projection, so the midpoint of a segment does not map to the midpoint of its image. A construction that finds midpoints with a ruler is finding the wrong points, by an amount that grows with the depth being foreshortened.

So the diagonal method is not a shortcut that avoids measuring. It is the only correct way to do it with the tools on the table, and measuring is the approximation. That inversion — the geometric construction exact, the measured one wrong — is worth stating because the intuition usually runs the other way.

Iterating a wrong map, for contrast #

The essay’s central claim is that iteration is safe because the map is exact, and the claim is only worth anything if the alternative is measurably unsafe. It is.

Take the by-eye repetition: guess each bay’s depth as a fixed fraction of the last. That is a map too — a scaling of the drawn depth — and it is not the projective map, so each application adds its own error to the accumulated one.

The wrong field measures the single-step version: the best by-eye method misplaces a post by metres on a scene a few metres deep. Iterated a dozen times, the row of posts is not slightly irregular; it is a row whose far end is in the wrong place by more than a bay.

Which gives the practical test, and it needs no theory: repeat the construction a dozen times and look at whether the last bay agrees with a bay projected directly. An exact construction passes at any length; an approximate one fails visibly within a few steps, and the failure grows rather than staying put.

Where else this map appears #

The bay repetition is a homology with the horizon as its axis, and that description matches three other constructions on this site — which is what makes the census worth having.

A pavement. Alberti’s construction lays a whole floor of squares, and the transversals it produces are the same geometric sequence this construction generates one bay at a time.

A shadow’s repetition. A row of identical objects lit by one lamp casts a row of shadows, and the map from one to the next is the same homology composed with the shadow’s own.

And an anamorph’s repetition. Marks laid out for one eye, repeated along the ground, repeat under the same map — which is why an anamorphic floor pattern can be extended without recomputing the eye.

Naming the shared object is not filing. It says that a result proved for one of them holds for all four, and the exactness measured here is the same exactness those constructions have.

Reading the sequence the bays make #

The bays form a geometric-looking sequence in the picture, and it is worth saying exactly what the sequence is, because the obvious guess is wrong in a way that matters for anyone eyeballing a drawing.

The depths in the world are equally spaced: one bay, two bays, three. Their images are at heights given by the perspective divide, so the drawn positions go as $1/(a + n b)$ — a harmonic sequence rather than a geometric one. Consecutive gaps therefore shrink, and the ratio between consecutive gaps tends to 1 as $n$ grows rather than settling at some fixed fraction.

Which is why a row of posts drawn by halving each gap looks wrong immediately and a row drawn by the diagonal method looks right at any length. Halving is a geometric sequence; the truth is harmonic; and the two disagree from the third post onward.

The cross-ratio is what makes the harmonic sequence checkable. Four consecutive bay edges have a cross-ratio of exactly 4/3 in the world and therefore exactly 4/3 in the picture — a number a reader can measure off any drawing of equally spaced things, and the fastest test there is of whether a drawing’s spacing is a projection of anything.

And the one thing it cannot do #

The construction continues a picture and never starts one, which is a limitation worth stating because it is easy to mistake for a shortcoming and is in fact the reason it is exact.

To draw the first bay a draughtsman needs something the picture does not contain: a real depth, in the world, related to a real width. That is metric information, and no straightedge construction supplies it — the measuring point brings it in from the eye’s distance, and Alberti’s distance point is the same number under another name.

So the division of labour is clean, and it is the same one the whole foundations field is organised around.

Metric information enters once, when the first bay is set, through a construction that knows where the eye is.

Everything after that is projective, and needs nothing but incidence.

A drawing is therefore correct in its proportions as soon as the first bay is right, and correct in its relation to a viewer only if that first bay was built from a real viewing distance. Those are two different kinds of correctness, and the repetition supplies the first without ever touching the second.

Why the manuals do not say any of this #

A closing note on the gap between the construction as taught and the construction as described here, because the gap is instructive rather than a complaint.

A manual gives the recipe and a figure, and the justification it offers — when it offers one — is that the diagonals of a rectangle cross at its centre, which is true of the rectangle and is not obviously true of its picture. The step from “true in the world” to “constructible in the picture” is the whole content, and it is a projective statement that the manuals predate or at least do not use.

What the projective account adds is not rigour for its own sake. It says which steps can be trusted at any length (the ones that are incidence), which need the eye (the ones that are metric), and what the failure of the by-eye alternatives is made of. Without it, all three look like the same kind of drawing advice, and a reader has no way to tell which will drift.

The one thing worth remembering #

A construction that iterates without drifting is doing something different from a construction that is merely accurate, and the difference is visible in a single test: run it a dozen times and see whether the error grows.

Here it does not — 8e-13 px at eight bays, 1e-12 px at twelve — because each step is an exact projective map and the composition of exact maps is exact. A construction that approximated would compound: twelve applications of a map that is 1% wrong is a picture 13% wrong, and by then the row of posts is visibly leaning.

That test is cheap and it separates the two kinds of drawing rule better than any amount of reasoning about them. The wrong field’s constructions fail it immediately; this one passes it at any length the paper allows.