Drawing convention — where it appears
Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.
Three distances in one landscape
A landscape assembled from a low station for the near ground, a level one for the middle and a high one for the far gives its furthest band 6.9 times the picture one camera would allow it — because the image of a fixed depth interval falls as one over depth squared, and its own station gives that back. What it costs is a jump in the rate at which depth runs, 2.50 at the first join and 2.75 at the second, and three views from one height leave no seam at all.
The same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
Size that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
A picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
The pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.
Bands with no overlap name a height for the eye
Read as a view, a stack of bands is one drawing of a family of eyes and row spacings. Keeping every figure clear of the row in front sets the least elevation: 40° for a procession and 74° for a crowd. There is a real parallel view that draws it exactly, an oblique one onto the wall itself, but it draws its people as paper. What the arrangement buys, on one wall, goes as the square of the number of bands.
Where a seam is allowed to go
Putting a join further off cuts the offset it forces from 210 px to 24 and leaves the turn at 23.2° exactly, so emptiness is a painter's only defence. And emptiness is not counted in metres: one twelve-metre object blocks 6.3 per cent of a landscape's depth wherever it stands and between 1.0 and 35.6 per cent of the picture, so how much room a seam has is set by where the landscape is crowded rather than by how much.
A 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.
A seam that bends trades a turn for a gap
A join laid along a river bank instead of across one depth was supposed to go round what crosses it instead of avoiding it. It can, at a price: the far band can be lifted by one amount only, so a seam that wanders in depth opens a gap between the bands — 3 px buys the first join 1.2 m of wander, and going all the way round a twelve-metre object costs 15.8 px. Across a hundred landscapes, bending opens under 0.4 per cent of the picture that a straight seam did not already have.
A 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.
A 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.
A 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.
A band lifted by columns bends its lines
A curved seam in a several-station landscape opens a gap because the far band is one picture lifted by one amount. Lift it column by column instead and the seam closes exactly — and the gap goes into the band's lines. Shift each column and every road bends by twice the old gap, horizon and all. Draw each column from its own eye and a road bends by 8.1 px near the seam and 3.0 at the band's far edge, dying only as one over its depth; the horizon stays straight; and the next join inherits 37 per cent of the gap the first one lost.
Two stations in one picture
A parallel floor under a perspective room found one map absorbing two centres into one sheet. Split a two-rule picture down the middle instead and each half hands back its own horizon — 7.80 px apart at a rule-mix of 0.02 — and no eye's position has anything to do with the gap, because an ordinary pinhole picture's recovered horizon does not depend on where the eye stood at all.
A stepped hand passes for a tiring one on a short strip
A painter who steps dividers from each row to the next leaves an error that accumulates — a walk from the edge begun at, with no fatigue in it. The fit of habit and creep takes about four-fifths of that walk before any reading of the scatter starts, because a walk is slow and so are the two numbers. What is left names the starting edge nearly as well as fatigue does, and on a six-bay strip it cannot say which of the two mechanisms left it: the verdict needs twenty-four bays on one strip, or sixteen six-bay strips by one hand.
A painter's closing can be read from one band
A band of a several-station landscape closed along a curved seam carries the record of how. A station per column bends a road by one over its depth, a column-wise shift bends every road alike, and one lift bends none. Four roads read to a pixel name the closing nearly every time, down to a station curve of about six centimetres, and the reading survives every depth being misjudged by the same factor.
A stepped hand's walk is in the creeps it was given
Fit each strip of a picture alone and a stepping hand's walk vanishes into a creep of that strip's own. Fit the picture once, with one habit and one creep for the hand, and the walks come back as a disagreement between strips that their small residuals cannot explain. Read against a line set once, eight six-bay strips name the hand ninety-five times in a hundred; the residuals of the same strips name it eighty-six, and needed sixteen strips for what the creeps do in eight.
One more habit takes two hands, and only part of a walk
A picture whose strips disagree was not painted by one steady hand, but the disagreement does not say whether a stepping hand or a second painter put it there. Refit the picture with a second habit and the answer is in how much of the disagreement that one number takes: nine-tenths and more for two hands, a share that shrinks with every strip added for a walk. Read against a line set on separate paintings, eight strips tell the two apart eighty-nine times in a hundred — and the split the second habit chooses is a map of who painted what, wrong mostly where an unknown starting edge lets one habit counterfeit the other.
A panel's creeps keep the order it was painted in, while the hand is steady
A painter who tires lets each pavement's gaps creep a little more than the last one's. Fit each pavement's creep from its transversals and rank them, and the ranking is the order the panel was painted in: with the creep growing a point a braccio from one pavement to the next and the hand at a pixel, the whole order of four pavements comes back fifty-seven times in a hundred, against one in twenty-four by chance. The step is the weak link — four short gaps fix its creep to two points — and a creep set by each pavement's length rather than by time reads as a confident order that is not the painting's at all.
A walk's order is on the floor, not on the page
A narrative panel whose copies are moments of one walk holds its order after all — in where the copies stand, because a walker moves a step between moments. Joined by the shortest route on the page, six copies seen from a standing eye come back in the order walked 68 times in 100. Undo the floor first, with the horizon the copies themselves supply and the angle of view guessed anywhere from half to twice the truth, and the same rule returns it every time the walk allows — leaving one bit, the direction, which a figure facing its way supplies.
Groups in a row are one painter twice; groups in turn say little
A second habit splits a divergent picture into two groups, and nothing in the split says whether they are two painters or one painter before and after a change. The order of painting can, because a hand that changed once leaves its groups in a row and two hands taking turns leave them alternating. On eight strips read to a tenth of a pixel, 93 panels in 100 by one hand who changed habit are read with their groups in a row and none by two hands in turn. But the change and the fatigue the order is read from are both drifts in time, so a change at one moment costs the rate of tiring three-quarters of its precision again — and a panel read as alternating has acquitted nobody.
Each copy's facing reads the walk past the place it doubles back
A walk that doubles back defeats every reading of a narrative panel from where its copies stand, because a later stop lands nearer an earlier one than a step. The way each copy faces is a short arrow on the floor, and a route that has to agree with the arrows as well as the distances follows the walk past the turn: on walks turning 70° between stops, facings read to 15° return the order 94 times in 100 where positions alone reach 59 — and say which end came first. But only once the arrows are undone onto the floor with the horizon the copies supply. Read straight off the page, an arrow pointing away from a standing eye is drawn nearly flat, and the same facings return 82.
A walker who stops is ordered by the way it turns
Three copies of a walker who pauses on one spot stand a few centimetres apart, and no reading of where they stand can order them. The way they face can — but not by the rule that followed a walk that keeps moving, which reads a three-copy pause turning 90° only a quarter of the time. A reader that ignores the direction of legs shorter than a step, and charges those legs instead for the turn between the copies' facings, reads it 92 times in 100 and costs a moving walk almost nothing. The middle copy of the pause is the hard one, and it is held only where the walker turned well clear of the error in reading each facing.
Nothing moves along the direction
A part slid four metres toward the reader along the direction an isometric drawing projects along keeps its drawn place to a ten-thousandth of a pixel, and the same slide seen from a station point moves it forty-four. An exploded drawing is not an approximation that works because the parts do not move far — it is an identity, and it is why cutaways are drawn in parallel systems.
Named alongside it
The objects these essays reach for when they reach for this one.
Identifiabilitydegrees of freedomHorizonResidualForeshorteningleast squaresModel errorStation pointFalsifiabilityFree parameterGround planeMoving viewpoint