The thread: Taught, and never measured — page 4
The circle in the square wants a number
Every manual draws a circle in perspective by inscribing it in a square, crossing the diagonals, and marking four more points about seven tenths of the way out. Seven tenths is right — along the diagonal of the real square. Along the drawn one it is 54 mm off a circle two and a half metres across, and the drawn diagonal is the only diagonal on the paper.
The other systemsThe dimetric the set square draws
An orthographic direction has two parameters and produces three axis scales, so the achievable triples are a surface rather than a list. The drawing office's dimetric — one axis at 1 in 8, the other at 7 in 8 — has the right three scales exactly and the wrong two angles, and the picture it makes is an oblique projection of a cube rather than an orthographic one.
Systems that kept the measureWhat survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
The rectangle behind the lensA close picture carries its own distance
A 50 mm lens focused at one metre stands 2.632 mm beyond its focal length, and a picture records where the lens stood. Recover that from two vanishing points, put in the engraved focal length, and the focus distance comes back — to 0.7 per cent at 1 m, 3.7 at 5 m, and as an interval 28 per cent wide at 30 m. A single picture has given a scale. What it has given is where the lens was focused, and the subject can be anywhere in a sharp band two to fifteen times wider.
Many pictures at onceA start needs the sign of its depths, not their size
Started part of the way from a courtyard toward its inside-out twin, a bundle adjustment returns to the truth from every start less than 42 per cent of the way — a start with a sixth of the true relief, the right way round — and falls into the twin from every start past 56. Between, neighbouring starts settle in different answers. The band sits in the same place at a 3° field, where the twin misfits by under a pixel, and at 25°, where it misfits by seven and a half.
Systems that kept the measureA grid stops a copy drifting
A figure copied by hand from the last copy drifts as a random walk, by the square root of the copies. Copied on a ruled grid, with each landmark brought back to a line, it does not drift at all, because rounding throws the error away. That is how a correcting code works. It holds only below a sharp threshold on the hand, and the grid's size is the trade between how much it corrects and how much it can say.
Constructing a viewThe line every nosing is on
Nothing in a staircase points up the pitch. Every surface in it is level or vertical, and its picture has a vanishing point off the horizon all the same — belonging to the line the front edges of the treads lie on. That point is not free: it is collinear with the travel point and the vertical point, and the angle it makes says the rise-to-run the builder chose.
Systems that kept the measureThe workshop that throws drawings away
Adding a rule that discards a drawing which looks wrong turns the mark copyist's spread from a growing random walk into a stationary process — the fitted exponent falls from 0.542 to 0.022, indistinguishable from the method copyist's 0.033 — and the two mechanisms then separate only by where they settle, 0.7002 against 0.8000, seven and a half spreads apart.
Many pictures at onceThe cameras decide where a narrow view settles
Scatter a narrow-field bundle adjustment's starting depths and the answer it reaches stops following them: of eighty scattered starts whose depths had the right sign, twenty-six fell into the inside-out twin, and of a hundred with the wrong sign, forty-three came home. Split the start in two and the reason is plain. With the cameras where they are, the courtyard comes home from its own inside-out points; with the cameras on the twin's side, it falls in from the true ones.
What each system gave upA tiring hand draws a different habit
A painter whose bays creep two per cent wider down a strip has drawn, to within a fiftieth of a pixel, what a steady painter with a different habit would draw — and on a weakly splayed floor the difference is the whole distance from a hand's even rows to a camera's log placement. One strip cannot tell fatigue from habit. A whole picture can, because a creep counterfeits a habit in proportion to the number of bays over the logarithm of the splay, and that is different on every strip.
The other systemsA circle off the coordinate planes
An ellipse template is cut at one ratio, and the ratio is the cosine of one angle: between a coordinate plane's normal and the direction of projection. A face tilted out of that plane needs a different ratio and — the half that gets drawn wrong even when the ratio is close — a major axis pointing somewhere else, perpendicular to the drawn normal rather than to any edge.
Constructing a viewThree-point, laid out with a straightedge
Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
Systems that kept the measureA grid on the wall is a scale without a projection
An Egyptian canon rules a wall into squares and counts a figure's height against the ruling — no horizon, no centre, and a length recovered to 1.4e-14% of error where the same reading taken off a pinhole misses by 58%. Applied instead to a pinhole picture, the furthest of six equal figures reads at 19% of its true height.
Mirrors that are not camerasCloser than they appear, by a factor with a number in it
A wing mirror of one metre radius held eighty centimetres from the eye reports 1.30 metres for an object at sixty-four, because the image of anything distant sits half a radius behind the glass. The size such an object subtends reads as a distance 2.60 times the true one, and the factor is exactly one plus twice the eye's distance over the radius — so the warning is a number, and it is larger for the mirror that is further away.
Many pictures at onceA scale bar is worth its ends, not its tape
A length measured at the start of a walk round a ring pins the walk's scale — but only to the precision with which the pictures place the length's two ends, 3.4 mm here, whatever the tape says. Measured to a thirtieth of a millimetre or to three millimetres, a one-metre bar leaves the worst camera at the same 89 mm. So a short length is not a long one measured well: a bar under about 13 cm tells the walk less about its scale than the walk already knows. And on an open walk no length at the start does better than 214 mm, because what is left there is drift, not scale.
What each system gave upA tiring panel keeps its order, not its direction
Let a painter's creep grow from one strip to the next as the panel is worked, and the strips' habits carry the order they were drawn in — but only as a line, never as a direction: tiring from the floor to the book and steadying from the book to the floor put the same drift on every strip. Four strips find the order a quarter of the time against a twelfth by chance; six find it nineteen times in twenty. And the order costs the reading its refusal: once it is free, two steady hands fit one tiring hand nearly as well as a tiring hand does.
The other systemsThe shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
Systems that kept the measureHigher on the page, and where that stops being true
A pinhole's image height above the horizon falls monotonically with depth over the whole 2–400 m range sampled here, and 57% of that whole range lands in the ground's last drawn tenth — the accumulation the oldest depth convention is quietly built from. Above eye level the ordering inverts, and at eye level exactly, five different depths draw one height, a spread of 0.0e+0 px.
Mirrors that are not camerasTwo people between mirrors see each other equally often
Put two people between a pair of mirrors and each sees some number of images of the other. The two numbers are always equal, and so is the apparent distance of each image against its partner — a path of light walked backwards is the same path. What is not shared is where each image appears, and what the shared count depends on turns out to be neither person's position but two quantities made of both: the difference of their angles about the mirrors' meeting line, and the sum.
What each system gave upA strip keeps its ratio, not the end it began
A painter dividing a strip into bays tires as they go, and each bay comes out a little larger than the last. Divide the strip from its far edge instead of its near one and the tiring runs the other way — but the rows record none of it: a strip divided from the far edge by a tiring hand is, to the last digits, a strip divided from the near edge by a steadying one. A whole picture recovers every strip's direction anyway, because one hand shared one rate of tiring across strips of different splay.
Mirrors that are not camerasA mirror pays for its field out of its aperture
Any map from the eye's angles to the world's is a mirror, by one line of arithmetic, so a mirror can be designed by what it does. The field is then a sum along the glass of one over the size factor, which caps it at the aperture over the smallest factor allowed, and a simple map reaches that cap. A surface free to take any profile still cannot read both directions at one constant factor unless it is flat.
Systems that kept the measureNo solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
What each system gave upA strip's scatter points to the end drawn last
The size of a tiring painter's bays cannot say which edge of a strip they started from, because a creep read backwards is a creep. The scatter of the bays can: a hand that worsens as it goes leaves its largest slips on the rows it drew last. One six-bay strip names its edge three times in four when the slip triples, a four-bay strip carries no evidence at all, and a workshop that divided every strip from one edge is named nineteen times in twenty from six strips — with no creep in the picture.
Many pictures at onceTwo short bars are not one long one
A long scale bar is awkward to carry, so the natural substitute is two short ones set far apart on the same wall. It does not work. Two 25 cm bars 3.5 metres apart leave the worst camera of a closed walk at 219 mm; one bar over the same span leaves it at 72. The gap between the bars is read by the pictures like any other distance, so it carries no scale — and the same holds for a long length measured as separate pieces. What does work is a tape read at every mark: pieces that share their ends add up to the whole, and pieces that do not, do not.