Free parameter — where it appears
Named by 40 essays across 14 fields — each of them below, with the objects they name alongside it.
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.
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.
Oblique is a shear, and the shear is the whole system
Cavalier and cabinet are usually introduced as easy perspective for people with a set square. They are not a simplification of anything — they are the answer to a demand no orthographic projection can meet, which is a front face at true size and a depth axis at full length at the same time. The two are locked on a unit circle, and buying both costs exactly 45° of obliquity.
A picture with two eyes in it
Several traditions draw the floor from one place and the people on it from another. No single camera produces both, as an earlier essay showed. What such a picture actually is has a measurement attached: give the rays their world points and ask for the one place they all pass through, and at a stride of separation the best answer misses them by six tenths of a metre.
Three distances in one landscape
A landscape assembled from a low station for the near ground, a level one for the middle and a high one for the far gives its furthest band 6.9 times the picture one camera would allow it — because the image of a fixed depth interval falls as one over depth squared, and its own station gives that back. What it costs is a jump in the rate at which depth runs, 2.50 at the first join and 2.75 at the second, and three views from one height leave no seam at all.
One parameter between two surfaces
Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.
The second eye is a shear
A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs.
The sharp band is a decision
One 50 mm lens at f/2.8 focused at three metres has a sharp band half a metre deep or an unbounded one, and nothing about the optics changes between them — only how large a blur disc a reader is prepared to ignore. Every quantity usually quoted about depth of field is that acceptance restated, including the rule that a third of the band lies in front, which is true at one distance and nowhere else.
A landscape that changes its rule halfway up
A reader handed only the marks of a three-station landscape recovers each band's own camera height without being told any of them — 1.60 m, 4.00 m, 11.00 m, to 3.6e-15 m. What that same reader cannot recover across a join is a common ground: the next band's own marks read as a ground point 2.40 m away from the true one at the first seam, 7.00 m at the second.
Counting the eyes needs the room
How many eyes made a picture is not a question the picture can be asked. Told what the room really measures, the rays refuse to meet and a second eye has been caught; told instead that the room is the one the picture is consistent with, the same rays meet exactly, at the first eye. The refusal is real and it belongs to the room.
Where the anamorph still works
A picture correct from one point raises an obvious question that nobody answers with a number — how far may the eye move. The answer here is exact rather than approximate: a wrong viewpoint composes the intended picture with a central collineation whose axis is the line the picture stands on, so the error is zero along that line and grows linearly upward, and a step sideways costs precisely as much as the same step upward.
Two grounds, and what the second one costs
The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.
The seats a screen will accept
Collect the seats whose picture is within a pixel of the one intended and the result is a solid — half a cubic centimetre in front of a curved desk monitor, a litre in front of a curved television. Ten times the tolerance is a thousand times the room, which is the pavement anamorph's own law arriving on an object that has nothing else in common with it.
The marks name the place, not the height
Run the camera-recovery round trip on an anamorph — hand it the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
A parallel floor under a perspective room
Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.
One picture and three people
A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.
An ambiguity is not an uncertainty
Eight marks to forty cuts a solid scene's pose error from 19.8° to 0.7° and leaves a flat one at 48°. The two failures look identical from inside — a confident answer, a residual at the floor — and they respond to opposite remedies, so telling them apart is worth more than either measurement.
The same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
A set cut for one eye
Build a colonnade four metres deep and cut every column so that its picture is the picture of one eighteen metres deep. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places. What gives it away is the second eye, and not by the ratio anybody would predict.
The stations are also a staircase
A level eye draws a ground point on a row that depends on the eye's height and the ground's only through their difference. So a landscape drawn from three stations at 1.6, 4 and 11 m over flat ground is, to 6 × 10⁻¹⁴ px across ninety-nine samples, the same picture as one eye at 11 m over ground stepped 9.4 m and 7.0 m above the far plain. The station heights a reader recovers are recovered on an assumption the marks neither supply nor test — and what would decide between the two readings is a cliff 103% of a band's own height, which is exactly where the convention lays its mist.
A seam breaks direction, not size
An eight-metre road crossing the first join of a three-station landscape is drawn 215.4 px wide under either band's rule, identically, and a six-metre post 161.5 px tall under either — the eye's height cancels out of any size taken at one depth. What does not cancel is where those sizes sit. The road's edges are turned 23.2° from each other and the post's foot lands 64.6 px out of place, and a painter butting two bands can absorb the offset and can never absorb the turn.
Where a seam is allowed to go
Putting a join further off cuts the offset it forces from 210 px to 24 and leaves the turn at 23.2° exactly, so emptiness is a painter's only defence. And emptiness is not counted in metres: one twelve-metre object blocks 6.3 per cent of a landscape's depth wherever it stands and between 1.0 and 35.6 per cent of the picture, so how much room a seam has is set by where the landscape is crowded rather than by how much.
What one oblique drawing shows
A parallel projection is a linear map from three dimensions to two, so it has a direction it throws away. Slide any point along that direction and its mark does not move — which makes every parallel drawing the drawing of a three-parameter family of solids, and the depth scale a convention chooses is one direction through the family rather than a boundary of it.
A seam that bends trades a turn for a gap
A join laid along a river bank instead of across one depth was supposed to go round what crosses it instead of avoiding it. It can, at a price: the far band can be lifted by one amount only, so a seam that wanders in depth opens a gap between the bands — 3 px buys the first join 1.2 m of wander, and going all the way round a twelve-metre object costs 15.8 px. Across a hundred landscapes, bending opens under 0.4 per cent of the picture that a straight seam did not already have.
The parallax you cannot shoot away
A stitched panorama's parallax has two halves and they do not behave alike. The one across the seam falls as the sine of half a frame spacing, so more frames buy it off; the one up the frame is the sine of half the frame's own height, and no quantity of shooting touches it. They cross at π over β, which has no pivot error in it at all.
A tilted target pays for its tilt in perspective
A calibration target of printed circles held square to the lens fixes a lens's first radial coefficient to 4.77 thousandths from ninety-six marks. Tilt it sixty degrees, fit the tilt along with everything else, and the same ninety-six marks fix it to 1.70 — nearly three times better. Squash the circles by the same angle without perspective and nothing is gained. What pays is the near half of each circle being drawn larger than the far half, which spreads a circle's marks across a band of distances from the lens's centre.
The ball stands at a focus
A ball on a floor casts an ellipse, and the point where the ball touches the floor is a focus of it — not near a focus, on one, to two parts in ten thousand million million. That is Dandelin's theorem arriving somewhere nobody puts it, and it is the only thing about the lamp that the shadow gives away.
The drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.
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.
Two stations in one picture
A parallel floor under a perspective room found one map absorbing two centres into one sheet. Split a two-rule picture down the middle instead and each half hands back its own horizon — 7.80 px apart at a rule-mix of 0.02 — and no eye's position has anything to do with the gap, because an ordinary pinhole picture's recovered horizon does not depend on where the eye stood at all.
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.
The eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
The room a divergent picture is a photograph of
A divergent construction depicts a rectangle on a plane leaning toward the camera, and the splay alone sets how far — 36.3° at a splay of 1.32. Stand a second such construction on the first one's far edge, as a wall, and the two recovered planes meet at 6.9°, not at the right angle a real room's corner would need.
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.
A drawn fold has a phantom
A Necker cube has two readings and so does a drawn fold, and the fold's second reading is not the supplement of the first. The drawing fixes each plate edge's component in the picture and leaves its component along the ray free up to a sign; a reflection identifies two of the four sign pairs, so there are exactly two plates — and a hundred-degree fold reads as eighty-seven as well.
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.
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.
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.
A turned cube punishes the eye, not the construction
Turn a two-point cube until one face is nearly square to the picture and one vanishing point runs 6,500 px off the sheet. The measuring point it swings to does not run with it — it stays on the sheet — and the constructed marks go on forgiving slips ten to three hundred times what the eye is allowed. What collapses is the by-eye placement: the far edge of the face that becomes a sliver must be put within a third of a pixel.
Named alongside it
The objects these essays reach for when they reach for this one.
Station pointDemonstrationDrawing systemForeshorteningHorizonVanishing pointConditioningcentre of projectiondegrees of freedomDepicted rectangleDepth compressionElevation