The thread: Taught, and never measured — page 5
The mirror with an eye is the one that is sharp
A convex mirror focuses a distant point at two depths, and the gap between them is exactly the gap between its two size factors, so a mirror's blur can be read off its map. Hold the two factors equal at every point and the blur vanishes. That mirror turns out to be a familiar shape, and it is also the only one with a single centre of projection.
Constructing a viewThe proportion is the assumption
Read the proportions of a rectangle out of a photograph of it and the answer is a function of where the centre of the picture is assumed to be. Sweeping that assumption across the horizon takes one drawn quadrilateral from one part in fourteen to slightly wider than square, every reconstruction a genuine rectangle, and only a fiftieth of the sweep within five per cent of the truth.
The other systemsThe ball a drawing does not draw round
An orthographic drawing of a sphere is a circle wherever the sphere is, and its centre is the image of the sphere's centre, exactly. A cavalier oblique drawing of the same sphere is an ellipse of aspect exactly √2 — and the drawing office reaches for a circle template. One formula covers both and the camera as well, and only the camera moves the centre.
What each system gave upA 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.
Many pictures at onceA length fixes the scale where it lies, and the worst camera nowhere
Move one measured metre round a ring of twenty-four cameras and the worst camera hardly notices: 199 to 225 millimetres on the open walk, 80 to 103 on the closed loop, wherever the bar lies. What moves is the scale. On the open walk a length is known to 0.41 per cent beside the bar and 1.58 per cent twenty walls away, and no longer bar at the start brings the far end below 1.53. A survey that must measure lengths everywhere on an open walk needs its references spread along it; one that closes its loop can put a single reference anywhere.
Drawn confidentlyA steady creep draws the rule's bow
The arc reading was built to find the constant-ratio rule by the shape its pavements bow the diagonal. A correct pavement painted by a hand that lets each transversal gap grow 4.5 per cent on the last bows it the same way — to 0.31 pixels, a cosine of 0.9994 — on eight braccia, and 1.5 per cent is enough on twenty-two. The arc reading measures how much a pavement's tiles lengthen, not what lengthened them, and against its counterfeit it is a coin. The rule's own definition still tells them apart: its gaps' logarithms are straight, a correct pavement's bend. Read to half a pixel, that names the rule on eight braccia 94 times in a hundred.
Constructing a viewThe quadrilateral no rectangle casts
The relation that reads a camera out of a drawn rectangle has a minus sign in it, and the minus sign is a refusal: two vanishing points on the same side of the assumed centre give the square root of a positive number, and no camera makes that quadrilateral out of a rectangle. Watching the refusal arrive shows what it is worth — one corner has to travel most of the picture's width before it fires.
What each system gave upA 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.
Drawn confidentlyA panel's pavements convict the creep one pavement hides
One crept pavement bows its diagonal exactly as the constant-ratio rule does. A whole interior has several — a floor, a step, a table's inlay, a loggia's coffers — and the rule bows each by the same share of its own bow, where one steady creep bows a twelve-braccio pavement 1.72 times as much and a four-braccio step 0.62 times: four pavements painted to a pixel name the rule against it every time. A creep tuned pavement by pavement to counterfeit every bow is caught instead by the gaps, whose curvature has one sign on every correct pavement: four pavements name the rule ninety times in a hundred at a pixel, eight pavements ninety-seven.
Light and mirrorsThe lamp comes out in rays and not in plan
One drawing of three posts and their shadows yields two points, and a curved floor treats them completely differently. The lines through each post's top and its shadow's tip meet at the lamp's image to a ten-thousandth of a pixel at every curvature, because a top and a tip are two points of one real ray. The lines through each foot and the same tips meet 113 pixels from the lamp's foot — and the lamp placed from an exact point and a wrong one lands 1.3 metres away.
Constructing a viewThree procedures, one panel
Alberti's lateral section, the distance-point construction and a pinhole camera put every transversal at the same pixel — and every reading of the finished drawing therefore returns the same number for all three. The methods are distinguishable on the desk and indistinguishable on the panel, which is the fact any attribution has to start from.
What each system gave upOne 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.
Mirrors that are not camerasA mirror's top edge waits for an eye above it
Between two upright mirrors a path of light from one eye to another, unfolded, climbs in a straight line from one eye's height to the other's, so every bounce lies between the two. While both eyes are inside the glass's height its top edge takes nothing at all. Above it, the images go in a fixed order — not the one with the most bounces, not the longest, but the one whose last bounce falls furthest along its path, in forty arrangements out of forty. The barber's corridor that sinks out of the glass is a mirror leaning a fraction of a degree, and its length goes as one over the square root of the lean.
Drawn confidentlyA 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.
The other systemsA drawn fold has a phantom
A Necker cube has two readings and so does a drawn fold, and the fold's second reading is not the supplement of the first. The drawing fixes each plate edge's component in the picture and leaves its component along the ray free up to a sign; a reflection identifies two of the four sign pairs, so there are exactly two plates — and a hundred-degree fold reads as eighty-seven as well.
Where to standThe stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
What each system gave upGroups 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.
Drawn confidentlyThe rims carry a part's taper, as finely as they are drawn
A turned part's two rim ellipses are enough to read its taper: the one axis that makes both of them circles lands on the true vanishing point, and looked at down that axis the rims give a one-degree taper back to the sixth decimal. Drawn by hand to half a pixel they give it to ±0.4°, a part pointing away from the eye to ±0.15° and one lying across the view to ±3°. The rims' sizes alone cannot do it, since a turn shrinks the far rim exactly as a taper does. And rims drawn with ellipse guides carry no taper at all — the guide's step reads as one.
The other systemsTwo outlines are two curves
The whole method of multiview drawing is the transfer line — a feature at a position in the front view is at the same position along that axis in the top view. On a flat-faced solid the feature is a vertex and the rule is exact. On a ball the two views draw two different great circles, meeting in exactly two points, and the transfer line joins places that are √2 radii apart.
Constructing a viewThe slip that leaves no trace
A distance point put twenty-four pixels wrong moves the pavement by two and a half and leaves the reader's projective test reading exactly four thirds. The same slip on Alberti's section moves the drawing by the same amount and is caught, so the difference is not the size of the error — it is that one of them lands back on the set of correct drawings.
What each system gave upTwo painters on their own clocks leave no record of their turns
If each of two painters tires with their own work rather than with the day, the habits a panel's strips report record each painter's own sequence and nothing of how the two sequences were interleaved: painted in turn or one after the other, the panel is the same to the last digit. What does carry the difference is the shape of the fatigue — one trend through the panel, or two that start fresh — and that shape names one hand or two painters 59 times in 60 at a tenth of a pixel. But only once the comparison is calibrated: taken at face value, the model that searches seventeen times as many orders wins by searching, and calls two painters one hand four times in five at a quarter of a pixel.
Drawn confidentlyA turned part drawn from the side keeps its taper in its sides
A turned part's two rims read its taper well when the part points away from the eye and lose it as the part turns across the view: drawn to half a pixel, a one-degree taper lying across the view reads ±3.1°. Its two sides do the opposite. Fitted together with the rims as one tapered part, the same drawing reads the taper to ±0.03° — a hundred times better — and a part seen from the side, the view it is usually drawn in, shows a taper of six hundredths of a degree.
What each system gave upA painter's rate of tiring signs every strip but the first
Two painters who tire on their own work at different rates leave the difference in a panel's rows: at a tenth of a pixel a gap of a point a strip is read 97 times in 100, though the gap itself is read only to within three-quarters of a point. Carried to a second panel by the same two, where they share one habit and nothing else tells their strips apart, the rates give a strip to its painter — every strip but each painter's first, which carries no rate at all, and every strip but those whose creeps coincide, which they do whenever one rate is a small whole-number multiple of the other. Painters at one and 1.7 points a strip are told apart on every later strip; at one and two, on 88 in 100.
Drawn confidentlyA turned part's outline reads its swelling from every side, and a straight taper hides it
A turned part that swells by 15% of its radius two fifths of the way along, drawn to half a pixel, gives up the swelling's height to ±0.0024 of its radius and its crest's place to ±0.003 of its length from one side-view drawing — and to about the same from every other view, square to the picture or lying across it. A swelling of one hundredth of the radius, under a pixel on the page, is read as 0.0098 ± 0.0022. Read as a straight taper, the earlier essays' reading, the same drawings put the swelling into the taper and read a 1° part as 2.2°.