What each system gave up
The picture whose lines spread
In a Byzantine icon the sides of a table diverge with depth. The standard account says the vanishing point is behind the viewer. It is not — it sits below the near edge, in front of the eye, and it is the vanishing point of a direction running down and away.
An inverse perspective is a leaning plane
Ask a divergent construction what solid it depicts and it answers: a rectangle, four right angles, near edge equal to far. What the splay encodes is not the shape but the plane's tilt — and a real square on a plane leaning toward the camera really does photograph with its far edge wider.
A centre and a measure are exclusive
Eight drawing systems, measured on five questions, with the pinhole as a row rather than the header. Exactly one has a centre of projection and it is exactly the one with no true measure — and loosening the measure test by a hair lets it in, which is what says the boundary is real.
Each system answers its own question
A comparison in which every system wins its own column proves nothing if the columns were chosen after the systems. The test that makes it a result is whether any system wins something it was not designed for — and two of them do.
What perspective gave up
The field ends by turning its own battery on the system it has been comparing everything against. Four quantities a pinhole destroys that the other systems keep, each measured by the same computation as the systems it is set against, and each the price of the one thing perspective has and they do not.
A yes in the table is a price
The comparison table says isometric, cavalier, the elevation and the plan oblique all keep measure. Priced on four hundred boxes, with each picture handed its own best ruler, the four charge 0%, 0%, 33.3% and 0% for an edge — and a pinhole charges 39.5%, only 6.2 points more than the elevation it is filed against. Turn the boxes and three of the four yeses cost something; only the plan oblique's stays free.
A page is bounded by a divide, not a centre
A pinhole draws the whole of an infinite ground in a bounded patch of page — each doubling of distance half the one before — while a handscroll spends the same page on every doubling and an isometric drawing spends three quarters of its page on the last one. It is tempting to credit the centre. A crossed-slits camera, whose rays miss any common point by 0.46 m, is bounded too: what does it is dividing by depth in both directions of the page.
Four surfaces, and no one camera that draws them
Read under one assumed camera, the floor, footstool, table and book of a constructed divergent picture imply tilts of 36.7°, 65.1°, 54.2° and 78.2°, where one camera photographing four parallel surfaces gives each of them 35.0°. But the spread between the tilts is 46.8° under a 260 px lens and 6.7° under a 5,000 px one, so it measures the lens as much as the picture. The measure that owes nothing to a lens is on the page: the nearest drawing one camera could make moves the far corners by 17.6 px.
A camera count needs a tolerance
Asked how few cameras could have drawn a splayed picture, a constructed stack answers four, three, two or one depending on how many pixels of redrawing a reader will allow — one camera at 17.6 px, two at 11.1, three at 4.6. The count is real: a picture built with two groups hidden in it gives back exactly those two, anywhere between nothing and 47.0 px. What it is not is a property of the picture alone, and the floor under it belongs to the hand — a one-camera drawing made by a hand that scatters its far corners by σ splits below about 0.57σ, where the 0.57 belongs to that hand and runs from 0.17 to 1.34 across a hundred and twenty of them.
One camera means one horizon, not one point
The test this field has been using asks whether a picture's surfaces share a meeting point. One camera photographing four parallel surfaces turned by different angles in their own planes gives them meeting points 1,065 px apart in column and identical in height to 3 × 10⁻¹² px — so the shared-point test charges 28.7 px to a picture one camera really took, and the charge grows with the turn. What one camera imposes is a shared vanishing line. The earlier verdicts survive intact, and for a narrower reason than they looked to have.
The tenth row has neither
A crossed-slits camera divides by depth in both page directions and its rays miss any common point by 0.354 m. Put on the comparison table it prices 38.8% on length, 69.9% on area and 58.2% on angle against a pinhole's 39.5%, 70.8% and 60.9%, and its midpoint drift is 15.6% — the pinhole's own figure. It keeps a true scale in no direction at all, and it bows a straight run of ground by 1.11 px, which no row with a centre does. Giving up the point buys nothing and costs a third thing besides.
The rows under a splay measure the bays, not the lean
A splayed strip drawn with its rows evenly spaced is still an exact picture of one flat plane leaning toward the eye — the straight sides fix the plane, whatever the rows do. What the rows decide is how deep each bay is, and evenly spaced ones make the near bay deeper than the far by nearly the square of the splay. That, and the six-per-cent gap between even rows and a camera's, depends on the splay alone — not on the lean, the distance or the lens.
The exclusion is two conditions, not ten rows
Ten systems have been measured and none has both a centre and a true measure. Ten is not a proof. Swept across the whole two-slit family — eighty-one members, both parameters run out to infinity — a centre appears at exactly the eight members whose slits sit at one finite distance, a measure at exactly the seventeen with a slit at infinity, and at none of them both. The case the two conditions appear to leave open is closed by the rays themselves: one divide is a slit, and eight page points use 6.02 m of it.
The rows count hands, not cameras
Rows drawn between two straight sides charge a camera nothing — every strip of a four-strip divergent picture reads back as a flat plane, whatever placed them, so a single viewpoint redraws the whole picture exactly. What they do fix is one number per strip, and that number survives the lens. Read the same drawing at focal lengths twenty to one apart and the lean runs from 36.4° to 86.0° while the habit stays at 1.000000000.
A vanishing line with a slope in it
Turn the plane about the view direction and no family of any surface's edges is level; each vanishing line acquires a slope, and a group must agree about two numbers rather than one. The count does not change character — a hand of four pixels costs the test 3.00 px at no slope and 3.27 at thirty-eight degrees of it. What the slope does expose is the redraw: holding each far edge at its drawn height charges 0.95 px to a picture one camera really took.
A tiring hand draws a different habit
A painter whose bays creep two per cent wider down a strip has drawn, to within a fiftieth of a pixel, what a steady painter with a different habit would draw — and on a weakly splayed floor the difference is the whole distance from a hand's even rows to a camera's log placement. One strip cannot tell fatigue from habit. A whole picture can, because a creep counterfeits a habit in proportion to the number of bays over the logarithm of the splay, and that is different on every strip.
A tiring panel keeps its order, not its direction
Let a painter's creep grow from one strip to the next as the panel is worked, and the strips' habits carry the order they were drawn in — but only as a line, never as a direction: tiring from the floor to the book and steadying from the book to the floor put the same drift on every strip. Four strips find the order a quarter of the time against a twelfth by chance; six find it nineteen times in twenty. And the order costs the reading its refusal: once it is free, two steady hands fit one tiring hand nearly as well as a tiring hand does.
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 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.