Every essay — page 20
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Drawn confidently
The constructions taught in every book on the subject, measured. Some are fine. The point is that nobody knows which until the drawing is asked what solid it depicts.
The rule that draws another room
The taught rule for spacing receding boards — each gap a fixed fraction of the last — is not a projection of anything, and it produces a pavement that is a correct perspective to within a fifth of a pixel. Of a room whose horizon is a hundred and seventy pixels from the one the panel drew. The error is not incoherence; it is a disagreement between two halves of one drawing.
Six tangents and the point nobody drew
Brianchon's theorem is a test a reader can run on a finished drawing with nothing but a straightedge — six tangents, three diagonals, and a question about whether they meet. Pointed at the drawing office's four-centre ellipse it rejects the curve by 1.7 per cent of the figure's own width, 546 times the instrument's own floor, with no true ellipse to compare against.
A mirror that is not parallel to the wall
Carry the depth in front of the glass an equal depth behind it, square to the wall. That is exact for a mirror hung parallel to the wall and 1.26 metres — 107 pixels — out for one turned 20°. Two invariants survive the turn instead, and one of the two nearly did not survive being tested, because it had been written in a form that could not fail.
The minor axis is not the axle
A wheel's perspective ellipse is supposed to have its short axis along the axle, and it does — on the principal ray, to 5e-14 degrees, and nowhere else. Off it the two part by 5.95 degrees on an ordinary frame while the drawn curves stay 0.98 px apart. A sphere obeys a rule of exactly the same shape and obeys it everywhere, which is why nobody caught the difference.
A cylinder has two different ends
The two end circles of a cylinder image as two different ellipses — 14.4° apart in the direction of their major axes and 0.918 against 0.839 in aspect — and the outline's straight sides touch neither of them where its major axis ends, missing by 28° round the ellipse and 35 pixels. Walking the cylinder away removes the difference between the ends and does not remove the offset of the touch, which settles at 7.3° off the principal ray and at nothing on it.
A tapered part meets at its apex
The sides of a turned part that narrows by one degree meet 54 pixels from the vanishing point of its axis, at the image of its apex, and a quarter of a degree already moves them 15. Drawn toward the vanishing point instead, a two-degree part loses nine tenths of its own taper. Flare it the other way by 5.9 degrees and a correct photograph prints its sides parallel; flare it further and they spread with depth.
A square plan is not a cube
An even-handed two-point cube is square in plan wherever its far edges go, and a cube at exactly one placement — 19.4 per cent of the way to each vanishing point on the layout measured. At the taught drawing's 42 per cent it is a square slab a third as tall as it is wide. Measuring points supply that placement, and they do not make a hand exact; they move its slip to marks where it costs a tenth as much.
The rule is exact for a floor that lengthens
The constant-ratio rule for spacing receding boards is an exact perspective — to the last digit, on the panel's own horizon — of a floor whose boards grow by the inverse of the ratio, 0.74 braccia deep at the front and 1.31 at the back on an eight-braccio pavement. The orthogonals agree with that floor. What says the tiles were meant to be square is a diagonal, which bends 8.1 pixels off straight where the reader's fitting test finds a sixth of one.
A turned cube punishes the eye, not the construction
Turn a two-point cube until one face is nearly square to the picture and one vanishing point runs 6,500 px off the sheet. The measuring point it swings to does not run with it — it stays on the sheet — and the constructed marks go on forgiving slips ten to three hundred times what the eye is allowed. What collapses is the by-eye placement: the far edge of the face that becomes a sliver must be put within a third of a pixel.
A straightedge convicts the rule on five braccia
The constant-ratio rule for spacing a pavement's boards leaves one mark a correct construction does not: its tile corners bow off the diagonal. The worry was that a painter's hand would bury the bow in its own scatter. At a pixel of scatter it does not, on any pavement of five braccia or more. What limits the test is not the hand but the page: a pavement drawn to one width bows no more at twenty braccia than at eight, while the hand's wander keeps growing with every tile.