Constructing a view

One, two and three point are one construction

The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.

The measuring point, and the step the method leaves out

Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose.

The horizon is at eye level — if the picture plane is vertical

The horizon cuts every standing figure at the same fraction of its height however far away it is, which is the most immediately usable fact in the subject. It holds when the camera is level, and a twelve-degree tilt is enough to spread the fractions by more than a percentage point.

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.

Brunelleschi drilled a hole in his panel

The first perspective demonstration in the European record came with its viewing point enforced — a hole through the back of the panel, a mirror held out in front, and one place to stand. That distance is computable from the panel's size and the angle the Baptistery subtends, and the answer lands squarely on the arrangement the account describes.

The plane is a choice

A projection has a centre and a surface, and they move independently. Keep the eye and turn the picture plane and every point of any scene lands where one 3×3 matrix says, to 2.5e-13 px. Move the eye instead and the matrix fitted to four points is exact at those four and out by 32.0 px everywhere else. The first is a homography of the picture; the second is parallax, and nothing about the picture can undo it.

Straightening does not move the eye

Correct a photograph's converging verticals and what comes out agrees with a level camera at the same point — one the correction was never shown — to 3e-13 px, with the verticals parallel to 0e+0°. The cross-ratio of four points along a ground line reads 1.3333 before and after, so the corrected picture measures exactly what the original measured, from exactly where the original was taken and nowhere else.

The horizon, and the fraction

The horizon crosses every upright at the point of it that stands at the camera's own eye height — always, whatever the picture plane is doing. It crosses at the same *fraction* of the drawn height only when the plane is vertical: tilt by 6° and the fractions spread by 0.08 percentage points, tilt by 4° and 0.06. One statement is an incidence and survives; the other is a ratio and does not.

The ramp has its own horizon

Every plane has a vanishing line, and a ramp's is not the ground's. Its uphill edges meet on a line above the horizon, and the angle at the eye between that meeting point and the same direction taken level is the gradient — 22.0000° recovered against 22° built, out of the picture alone, with no scale, no ruler and nothing known about the scene except that the ground is level.

The bay repeated by a straightedge

Draw one bay, then repeat it by diagonals alone — no measurement, no scale, no arithmetic — and after twelve bays the constructed corners sit 1e-12 px from the corners the camera projects. It is exact because the operation being iterated is a homology of the picture, not because the draughtsman was careful, and that separates it from every construction in this site's `wrong` field.

Carrying a height across the room

A known height at one place on the floor, and the same height wanted at another: two lines settle it, and they settle it exactly, at every camera and every pair of positions. What the recipes never mention is that one of those two lines has to be drawn to a point that is usually not on the paper — 3,300 canvas widths away in the case drawn here — and that the repair is not to extend it further.

A picture with nothing straight in it

Every construction on this site is handed the horizon, and a photograph of a crowd, a hillside or a curved façade has no straight edge to give it. What such a picture does have is repetition — and three things of one height put the horizon exactly where the camera has it, from the picture alone. Two things do not, and three standing abreast do not either, and both refusals are the reader's ordinary situation.

Seven is not a power of two

Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.

The circle in the square wants a number

Every manual draws a circle in perspective by inscribing it in a square, crossing the diagonals, and marking four more points about seven tenths of the way out. Seven tenths is right — along the diagonal of the real square. Along the drawn one it is 54 mm off a circle two and a half metres across, and the drawn diagonal is the only diagonal on the paper.

The line every nosing is on

Nothing in a staircase points up the pitch. Every surface in it is level or vertical, and its picture has a vanishing point off the horizon all the same — belonging to the line the front edges of the treads lie on. That point is not free: it is collinear with the travel point and the vertical point, and the angle it makes says the rise-to-run the builder chose.

Three-point, laid out with a straightedge

Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.

A drawing has three horizons

The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.

The arc every eye stands on

Four drawn corners known to be a rectangle fix the horizon of their plane and nothing else. The eye that drew them has to see the two vanishing points at a right angle, so it lies on the circle those points are a diameter of — and every point of that arc reconstructs a genuine rectangle, with right angles to five parts in ten million million of a degree, and a different proportion.

The proportion is the assumption

Read the proportions of a rectangle out of a photograph of it and the answer is a function of where the centre of the picture is assumed to be. Sweeping that assumption across the horizon takes one drawn quadrilateral from one part in fourteen to slightly wider than square, every reconstruction a genuine rectangle, and only a fiftieth of the sweep within five per cent of the truth.

The quadrilateral no rectangle casts

The relation that reads a camera out of a drawn rectangle has a minus sign in it, and the minus sign is a refusal: two vanishing points on the same side of the assumed centre give the square root of a positive number, and no camera makes that quadrilateral out of a rectangle. Watching the refusal arrive shows what it is worth — one corner has to travel most of the picture's width before it fires.

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.

What a panel says about its maker

The reading assembled over this row, run against every procedure sixty times and scored — with the failures reported as carefully as the successes, because three of the five rows are refusals. A drawing names the class of error in it, not the recipe that produced it, and one procedure it never names at all.

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.

The 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.

What 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.

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 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.

A 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.

Figures on a street that slopes

Equal-height figures have their heads on one line, and the taught rule says the line is the horizon. On a street rising at 8.33 per cent the heads are still collinear to 2.8e-14 pixels and the line is 58.9 pixels above the horizon — the street plane's own vanishing line. The taught rule loses 1.46 m of a 1.62 m figure at the far figure, and runs out of figure altogether at 19.4 m.

The centre of the picture is not the centre of the paper

A crop translates the image rectangle, so the picture's optical centre leaves the middle of the sheet and the focal length does not move — 81.3 pixels apart at a fifth of the picture, with the horizon at 62.5 per cent of the print. A reader who takes the paper's middle for the picture's stands 2.36 cm out of position, which is 4.9 degrees of the wrong direction.

Copying 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.

The hook is the centre, and the eye is not

Dürer's string frame projects from a ring of iron driven into a wall, so its centre of projection has a position anybody can measure with a tape. A hook 120 centimetres behind a frame 56 centimetres wide makes a drawing correct from 34.3 centimetres shown 160 millimetres across, with its principal point 161 px off the middle of the sheet. The gridded veil projects from a head instead, and 30 millimetres of head moves its marks by 13.27 millimetres.