Transversal — where it appears
Named by 22 essays across 6 fields — each of them below, with the objects they name alongside it.
Alberti draws a pavement, and chooses where the reader stands
The costruzione legittima of 1435 is exact. Run as a construction — with drawn rays and drawn intersections rather than the formula it turns out to satisfy — it agrees with a pinhole camera to six parts in a hundred trillion of a pixel. And it has one free parameter that the recipe never names, which is the distance from the eye to the panel.
The distance point is the viewing distance, drawn
There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.
A texture does not interpolate on the page
Walking across a drawn surface at a constant rate walks across the real one at a rate that changes, and the worst gap is a closed form in the depth ratio alone — 0.52 at ten to one, more than half the whole range. It is exactly the error a person makes dividing depth by eye, made by a machine, and the fix is the fourth coordinate the pipeline kept.
The diagonals find the middle
Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.
The lens a pavement can hide
A photographed pavement reads as a correct drawing up to a radial coefficient of about four tenths — a lens strong enough to bow a straight edge across the page by nearly six pixels and to print as twenty per cent distortion at the frame's corner. The reason is that a pavement sits near the principal point, which is the one part of the frame a radial map barely touches.
Four lines have a cross-ratio
The cross-ratio is introduced as a property of four points on a line. Its dual — four lines through a point carry the same number, and any transversal reads it back — is not a second theorem. It is why the first one is true: four rays from an eye are a pencil, every picture of them is a section of that pencil, and a quantity belonging to the pencil cannot depend on which section was taken.
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.
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.
Along a line of constant depth the page is affine
Stepping a texture by a constant amount per pixel is wrong across a receding floor and exactly right along any line of it that stays at one depth — and on every plane those lines run parallel to its own vanishing line. Turn a 120 px span 1° away from that direction and it is 0.79 px out; roll the camera a hundredth of a degree and a floor drawn to 30 m is out by 0.69 px on its worst scanline.
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 tilted span walks a staircase
A span along a banked floor's constant-depth direction is exact, and a renderer visits pixels rather than the span. Snapped to the grid, a 120 px span at a 20° bank costs 0.577 px where the same span along a page row costs 13.26 — twenty-three times better — and it never rises above 1.22 px at any bank. The price is bookkeeping: a band of twenty-four such spans draws 53 of its 1,368 pixels twice.
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.
Three 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.
Four 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.
The 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.
One 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.
Stepped, or measured from the zero
The same hand at the same precision, laying the same braccia off two ways — dividers walked from the last mark accumulate and grow as the square root of the count, while marks set from a common origin do not accumulate at all. The difference is not which method was used — it is where the zero is, and only the second is recorded in the drawing.
Dividing 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.
Two rules for one pavement
Vignola set out two rules for laying a tiled floor and asserted that they agree. Executed from the same ground line and the same free parameter they agree to 2e-13 px; executed from the numbers their own wordings invite they part by 24 px. The quantity that separates them is the distance the reader has to stand at, and neither rule names it.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Projective invariantResidualCross-ratioHorizonViewing distanceAttributionDistance pointFalsifiabilityMeasuring pointModel errorProcedureTolerance