Every essay — page 24
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
Where to stand
A perspective picture is correct from exactly one point in the room. Every figure here computes where that point is, which turns the oldest complaint about wide-angle photographs into arithmetic.
A fold names the height
A pavement anamorph’s marks fix where the reader must stand and leave how tall they are entirely free — every height explains the marks exactly, to the last bit. Put one crease in the floor and the freedom is gone, because two degrees of fold makes a ten-centimetre error in the height leave six tenths of a millimetre, and a right angle makes it nine.
A flight that has ends
A real staircase has three families of faces and the rule proposed last round predicted three pictures. It supports four, because a picture is fixed by a direction and the flight's two sides point opposite ways — one kind of face that no single direction reaches both of. The rule counts words and the object counts orientations.
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 cube that is a box
The two-point cube every book teaches has a step it supplies no construction for. Place the two far edges symmetrically and the drawing depicts a square plan for free; place them eight points apart — a difference invisible on the page — and it depicts a box 1.4 times shallower than it is wide.
Dividing depth by eye
Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.
The sixty-degree cone of vision
Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.
The third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.
Both vanishing points on the paper
Putting the two vanishing points on the sheet is presented as a composition rule. It is a statement about the reader: with the two points one page-width apart the picture is a 90° view, correct from 80 mm, and a reader holding it at arm's length is shown a room five times as deep as the one drawn. The layout that is honest at arm's length puts both points four and a half pages off the sheet.
Which way the drawn circle leans
Two rules are given for the direction a drawn circle's short axis runs in: along the axis of the cylinder, and pointing at the centre of vision. On the optical axis both are exactly right. At the edge of an ordinary frame the first is out by three and a half degrees and the second by seventy-nine. The statement neither of them is comes out of differencing the projection, and it matches the drawn ellipse to a hundredth of a degree.
The forty-five degree shadow
Draw the shadow at forty-five degrees and make it as long as the object is tall. In the plan that is exactly a sun halfway up the sky. Applied on the paper it puts four posts under four different suns — altitudes from sixteen to twenty-nine degrees, azimuths thirty-two degrees apart, and shadows between one and three-quarters and three and a half times the height. No drawing angle brings them together.
Measured down from the waterline
Whatever stands so far above the water, draw its reflection the same distance below. Through a vertical picture plane that is not an approximation — it is the reflection, to the arithmetic floor. Tilt the camera twenty-two degrees and it is eleven pixels out. Draw a gull, which touches the water nowhere, and guessing its waterline point wrong by two metres of depth costs fifty.
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 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.
A 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.
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.
The 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.
A 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.
A 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°.
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.