Every perspective picture is correct from exactly one point in the room.
It is not a figure of speech. A picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm — which is the whole of why wide photographs are said to distort faces. They do not. They are being looked at from four times too far away. Every figure here is projected from a camera with a focal length and states the distance it is correct from, and the camera is recovered from the picture afterwards to prove the picture is a projection of anything at all.
Just added
8 essays · 20 September 2026 · everything, newest first
A shadow decides which landscape it is
One sun over a several-station landscape is one sun: every band images the light's direction and its shadows' at the same page point, and each reports the altitude as 22.000000°. A shadow is still dislocated at the join, and the reading that could not be settled by any ground point is settled by one — because a ray crossing the seam has a riser to descend that the flat reading does not give it, and the two tips land 5.94 m and 11.7 px apart.
What each system gave upThe 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.
What a machine computesA 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.
What each system gave upA 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.
Start anywhere
the most recent from each of 20 fields · all 434 essays
The theorem that is obvious one dimension up
Desargues in the plane needs a proof and in space needs none — two triangles in different planes have their corresponding sides meeting on the line where the planes cross, and the meets land 1.0e-14 metres off it. The plane figure is a shadow of the spatial one, and five different solids cast the same photograph to 2.0e-12 pixels.
Constructing a viewThe 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.
Where to standA flight that has ends
A real staircase has three families of faces and the rule proposed last round predicted three pictures. It supports four, because a picture is fixed by a direction and the flight's two sides point opposite ways — one kind of face that no single direction reaches both of. The rule counts words and the object counts orientations.
Surfaces that are not flatA projector that is not at the dome's centre
A projector 0.40 of a dome's radius off centre puts its own picture up to 23.6 degrees from where it belongs, and the pre-warp that corrects it is exact for one seat and only one. Two metres from that seat costs 11.5 degrees of the same displacement, wherever the projector itself stands — because the correction never knew where the projector was in the first place.
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.
Light and mirrorsThe lamp and the floor cannot both be recovered
Every member of a one-parameter family of lamp-and-floor pairs draws the identical photograph to a fraction of a thousandth of a pixel — one member swings the lamp 90 cm and tilts the floor 4.3° and the picture does not move at all. A tape measure or a post of known height each pin the true member uniquely, and so does a wall known in advance to be square to the floor, whose angle to the recovered floor peaks at exactly 90° at μ = 1.000 and nowhere else; a receiver merely known to be parallel drifts thirteen times more weakly.
Mirrors that are not camerasCloser than they appear, by a factor with a number in it
A wing mirror of one metre radius held eighty centimetres from the eye reports 1.30 metres for an object at sixty-four, because the image of anything distant sits half a radius behind the glass. The size such an object subtends reads as a distance 2.60 times the true one, and the factor is exactly one plus twice the eye's distance over the radius — so the warning is a number, and it is larger for the mirror that is further away.
Measuring from one pictureWhere each closure enters the stratification
The five facts that turn a photograph's ratios into metres do not all do the same job. Read on four quantities through the map each supplies — a cross-ratio, a ratio along a line, an angle, a length — three of them return all four exactly and two return only the first three. Set the free scale ten per cent wrong and the whole ten per cent appears in the length and nothing appears in the other three, to thirteen digits.
Through water and glassThe residual does not warn
Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.
The real instrumentNo design separates the two coefficients
Separating a squared term from a fourth-power one was supposed to need marks at radii far apart, which is a statement about where edges are placed. It is not: one straight edge already runs from 20 px to 326. Spreading ninety-six marks over three edges or sixteen changes the answer by 23 per cent, spreading the offsets makes it 19 per cent worse, and the correlation stays at −0.98 whatever is done. What a plumb-line calibration determines is one number, to 1.52 thousandths, and which number depends on the model.
The second eyeSeeing the scene fences in the plane at infinity
A reconstruction made without the calibration does not know which of its planes is infinitely far away. Requiring every point to lie in front of both cameras rules out every candidate that would tear the courtyard, and what is left is a convex region — 32 per cent of a generous slice — that always holds the true plane and never shrinks to it.
What a pair is forThe second disparity cuts cells
A point off the plane of the eyes has a vertical disparity as well as a horizontal one, and quantising both cuts a room into 7,663 cells where one coordinate cut 179 shells. The gain is entirely vergence's — two eyes looking straight ahead have no vertical disparity at all, exactly — and it is largest where the first reading is already finest: 60.8 in the near metre and 3.7 in the far band.
Many pictures at onceA start needs the sign of its depths, not their size
Started part of the way from a courtyard toward its inside-out twin, a bundle adjustment returns to the truth from every start less than 42 per cent of the way — a start with a sixth of the true relief, the right way round — and falls into the twin from every start past 56. Between, neighbouring starts settle in different answers. The band sits in the same place at a 3° field, where the twin misfits by under a pixel, and at 25°, where it misfits by seven and a half.
The eye that movesWhere 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.
Systems that kept the measureWhat a removed wall costs that a removed roof does not
Fitting a single centre to a building with its near wall deleted lands at 3.0e-15 m — the arithmetic floor — because deleting a wall does not touch the projection, only which surfaces are drawn. Fitting the identical routine to the same building with its roof removed does not return a number at all: handed a bundle of genuinely parallel rays, it refuses outright.
What each system gave upA 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.
What a machine computesA 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.
The rectangle behind the lensA close picture carries its own distance
A 50 mm lens focused at one metre stands 2.632 mm beyond its focal length, and a picture records where the lens stood. Recover that from two vanishing points, put in the engraved focal length, and the focus distance comes back — to 0.7 per cent at 1 m, 3.7 at 5 m, and as an interval 28 per cent wide at 30 m. A single picture has given a scale. What it has given is where the lens was focused, and the subject can be anywhere in a sharp band two to fifteen times wider.
The second projectionA stereo picture is drawn for a level head
Every stereo pair is drawn for two eyes level with each other — a point's two images share a row and differ only across it. Tilt the head 10° in front of a television and that difference turns partly vertical, 14.46 arcminutes for anything drawn at infinity, and the two sightlines to a point stop meeting. At a desk monitor the same fifteen-arcminute limit arrives at 2.58°.
Drawn confidentlyThe 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.
The longest series
an idea, and the distinct arguments standing against it · all 193 series
Anamorph · 8 Divergence · 8 Pushbroom · 8 Exclusion · 7 Distortion · 6 Threedistances · 6 Aperture · 5 Disparity · 5
Threads running through
themes, not chapters
Projected, not constructed
Every point in every figure here comes out of a camera with a focal length, not out of a line run to a vanishing point that was placed where it looked right. A picture no camera could produce fails to build rather than appearing with nothing to say so.
The viewer is in the geometry
A perspective picture is a projection from a point, and that point is a fact about the picture — computable from its focal length and the width it is shown at. Every figure states the distance it is correct from, because leaving it out is what makes the subject feel like a matter of taste.
The round trip
Draw the picture from a known camera, forget the camera, recover it from the drawn edges alone, and compare. Agreement to fifteen digits is a statement about the geometry; anything less is a statement about the drawing.
What a projection destroys
Length goes, angle goes, area goes, and the ratio in which a point divides a segment goes. Knowing precisely what is lost is what makes the one surviving quantity worth anything, so the losses are measured alongside it rather than mentioned.
One machine used twice
A shadow is a projection from the lamp. A reflection is the view from a camera on the far side of the mirror. A parallel drawing is a photograph from infinitely far away. Three subjects usually taught as three recipes are one operation with the centre moved.
The surface is a choice
A picture has to be cast onto something, and the something is decided rather than given. Flat keeps straight lines straight and cannot reach 180°; a cylinder reaches all the way round and bows every horizontal; stereographic keeps every angle and no scale. The price of each is a number, and the corner where a surface pays nothing is empty.
One picture, one length
A photograph gives up every ratio in the scene and no size at all: scale the world and the eye together and not a pixel moves. So a measurement from a single view is a ratio until the reader supplies one length, and which length that is decides how good the answer can be.
Taught, and never measured
The standard constructions are drawn here exactly as they are taught, and then asked what solid or what spacing they depict. Some answers are fine. The point is that the method itself supplies no way to find out, so nobody drawing knows which case they are in.
Not a projection at all
Every theorem quoted on this site is a theorem about a map through a centre. Water is not one; a real lens is not one. Rather than mention that and move on, these essays measure what survives and what does not — a cross-ratio 1.2% out where the pinhole is exact to the last bit, a bundle of rays that misses its own centre by ten millimetres, and the two things that survive anyway.
Fitted, not assumed
A distortion coefficient recovered from bent lines with no calibration target. A principal point computed instead of guessed to be the middle of the frame. A pivot offset read off the parallax it leaves behind. Each is the site's round trip pointed at a quantity that is usually filled in from a manual, and each comes with the conditioning number that says how well the data actually determined it.
What a second picture adds
One view fixes a ray and a set of ratios. A second fixes a point, the whole three-dimensional shape, and where the other eye was — recovered from the two drawings and nothing else. It does not fix a size, and it does not fix itself when the eyes are close together. Each of those is measured here rather than stated.
Determined up to something
Every recovery on this site returns an answer with a qualifier attached. One picture gives ratios and no size. Two give shape and no size. Many give a scene and a camera track up to seven numbers no quantity of pictures supplies. The qualifier is not a shortcoming to apologise for; it is the exact statement of what the pictures contain, and here it is counted, walked along and shown to be free rather than merely small.
Projected twice
A photograph is a projection onto a sensor and then a projection from a screen into a room, and the second one has a correct point exactly as the first does. Every essay before this quoted its viewing distance against an assumed figure width; here the assumption is replaced by a chain — focal length, sensor width, display width — and the two projections are compared. Almost nobody is standing where the picture says to, and what that does is already measured.
Made by something finite
The departures in these fields are not optical. Depth is kept in a fixed number of bits, the picture is a grid of samples that need not be square, and the frame is read over an interval and mostly a row at a time. None of that changes what a projection is; all of it changes what this particular picture is a projection of, and each one has a closed form and a control.