The thread: Projected, not constructed — page 6
A line is a space of its own
Most of what is said about projective geometry in pictures is said about the plane. One dimension down there is a smaller object with a complete theory: a point of a line is one ratio, a map of a line is three numbers, three pairs fix it, and the cross-ratio is not merely an invariant but the only one — which is a claim that can be made to fail.
Constructing a viewFour marks before anything is said
A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.
What a machine computesJittered frames rebuild a camera's horizon, saw-tooth and all
A renderer that moves its sample a fraction of a pixel every frame and averages the frames turns each pixel's single sample into an average over the pixel's area — which is what a camera's pixel records. Along a hazy, rolled horizon that removes the quarter of the shimmer point sampling adds, 0.082 a column down to 0.061, and stops there: eight Halton frames are within a hundredth of the camera, and sixty-four are no better. The rest of the saw-tooth is the camera's own, and so is the horizon's step. Only a band of repainted ground goes below them, and jitter's own price is a flicker of 0.29 a frame until a history averages it away.
The second projectionTwo walkers share a layout drawn for the worse of them
A projector turned to its wall can re-lay its corrected picture for one walking viewer and give them up to thirty per cent more detail per degree on the projector's side. For two, it draws the layout that serves the worse-served best. Walking the same way a metre apart, the one nearer the picture's centre keeps 99 per cent of its own gain and the other 77, falling to 44 at four metres apart. Crossing, they keep two per cent. A viewer on the far side vetoes the other only while it is the worse-served — and at different depths the shared layout takes from the walker it serves better, leaving them below the even picture.
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.
What survivesThe map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
What a machine computesA rolling horizon outruns the history that was meant to quiet it
Jittered frames averaged in a history draw a still hazy horizon as a camera does, four times nearer than one centred sample. Roll the camera and the history has to follow. Left in place, it is worse than no anti-aliasing at all from a degree a second. Reprojected exactly through the roll, it still reads last frame's picture between pixels, and an interpolated step is a softened step, softened again every frame: past one and a half degrees a second the history sits further from the camera than a single centred sample, and at sixteen it paints the first row of ground as sky. No blend fixes both its error and its pops. Four rows of band move with the horizon popping a fortieth as much — at the price of a quarter of the fog's range in repainted ground.
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.
Constructing a viewOne hand step each
Every classical perspective construction has exactly one step a person performs by hand, and the four constructions perform four different steps. That single difference decides everything a finished drawing can say about its maker, because the answers are identical and only the mistakes are not.
Surfaces that are not flatThe horizon's shape belongs to the surface
The horizon is one great circle of directions whatever draws it, and at zero tilt all six named surfaces draw it straight. Tilt the camera and they separate — and the cylinder, not the equirectangular surface, is the one whose horizon is exactly a cosine, to 9e-16 against 8.3e-3.
What a machine computesA clamp keeps the sky clean and leaves the rolling horizon smeared
Clamping a jittered history to the range of each pixel's neighbours is the renderer's standard guard against a moving edge smearing, and on a rolling horizon it does half the job. Two rows from the horizon, where the neighbours agree, it pulls the smear back at once: a sky row the history had darkened to 0.88 is restored to 0.99. On the rows the horizon crosses, the nine neighbours hold sky and ground together, the window runs from 0.12 to 1.00, and the smear passes through it untouched. The crossing with a plain centred sample moves from 1.5° a second of roll to 1.8; at 16° a second the history is 0.070 from a camera instead of 0.093, still nearly three times a centred sample's 0.025. And a clip tight enough to reach the smear brings the jitter back: on a still camera it pops four times as hard.
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°.
The other systemsNothing 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.
The other systemsTwo distances to infinity
A parallel projection is the limit of a perspective one, and the limit arrives twice. Which faces get painted settles within three object radii, because a face is either round the back or it is not; where each mark lands falls like one over the distance and is still out at thirty-four. Far enough away has two answers an order of magnitude apart.
Constructing a viewDividing to a point off the board
A wall turned forty degrees to the view has its vanishing point 0.65 canvas widths past the edge of the paper, and the construction that aims every course at it without ever reaching it is exact to 1e-13 px. Putting the vertex where the sheet ends instead costs 20.9 px, which on this wall is 300 mm of masonry, and nothing in the drawing says so.
Constructing a viewThe bays that are not equal
The diagonal repeats a bay exactly and forever, and what it reaches is the whole multiples of that bay and nothing else. An arcade of 1:1.5:2 is 250 mm short of its boundary with no halving, exact after one, and an arcade of 1:1.333:2 is out of reach at every depth whatever. A measuring point lays out all three to 6e-14 px.
Constructing a viewWhat a straightedge reaches on a receding line
Three marks on a receding line fix a coordinate, and everything a straightedge builds from them is a ratio of whole numbers. Three complete quadrangles put 1655 marks on the unit interval with no gap above 0.0038, a third costs two quadrangles where repeated halving never reaches one at all, and the same net built on a line that does not recede arrives at the identical 30 coordinates.
What survivesA point and a line are one object
Every projective statement has a partner got by swapping the words point and line, and the partner is true whenever the original is. Run on this collection's own Desargues configuration, all thirty of its incidences survive the exchange to 5 × 10⁻¹⁶ — and three of the dual's ten points land at infinity, which is a fact about where the drawing sits on the page rather than about the theorem.
Constructing a viewThe stair that turns has a vanishing point that moves
A spiral stair's treads are one rectangle turned by a constant angle, so every front edge has its own vanishing point and the twelve lie on one horizon to 2.8e-14 pixels. Four consecutive of them read a cross-ratio of 1.353720, and the formula behind it holds no focal length, no principal point and no eye — so a photograph gives the builder's 12° turn back.
Constructing a viewA measuring point for a ramp
Stepping true distances along a receding line needs a measuring point, and every printed rule puts it on the horizon. On a 1 in 6.0 ramp the ramp's own point lands every tread to 1.2e-13 pixels and the ground's puts the sixth one-metre tread at 2.57 metres instead of six. A halfway construction separates the two halves of the mistake, and the wrong radius costs 0.083 metres of the 3.43.
What survivesDesargues read the other way
The theorem's converse is not a second theorem. Exchange points and lines in the ten-point configuration and every one of its thirty incidences holds to 5 × 10⁻¹⁶, the axis becomes a point, the three axis points become lines through it — and what has been written down is the converse, read off the drawing rather than proved.
What survivesThe quadrilateral that finds the middle
The harmonic conjugate is usually built from four points. Built instead from four lines — the dual construction, the same number of straightedge steps — it lands on the same fourth point to 1.2 × 10⁻¹⁵ of the range's own length, gives a cross-ratio of exactly −1, and refuses the midpoint, whose conjugate is at infinity.
What survivesFour points on a conic look the same from anywhere on it
Four marks on a photographed circle subtend the same cross-ratio at every point of the curve — 1.627695, unmoved over twenty-two positions of the fifth point, to 1.1e-13 degrees of projective spread. A fifth point 6.1 pixels off the conic reads anything from 1.00 to 2.52, so the invariant belongs to the curve rather than to the four marks.
Constructing a viewCopying square by square
The taught grid workflow sets a pavement's cell corners out exactly and then fills each cell by eye, and the corners are right while the fill is not — 3.30 px on a picture 690 across at eight cells, falling as the square of the cell. On a wall square to the camera the same fill reads 3e-13 px, which is why the method feels reliable.