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.
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20 essays · 1 November 2026 · everything, newest first
The horizon is in the window and almost unlit
Snell's window holds the whole sky in a cone, right down to the horizon. Geometry says so and it is true. The surface passes less and less of that light towards the rim, so the last five degrees of sky get a quarter of a degree of the disc at under a third of their brightness. The window's famous edge, which looks like the surface's doing, depends on what is on the bottom, and over a white pool it disappears.
The rectangle behind the lensA box leaves a line of cameras
Let pixels be any width and one picture of a box has three equations for four unknowns, so it does not read one wrong camera but a whole line of right ones. The principal point sweeps across the frame as the width is assumed. Every member keeps all three right angles exactly, and the line has ends, because a squeezed triangle can stop being acute.
Many pictures at onceA longer lens buys parallax and not the direction
Walk a pair away from a scene and the parallax a homography cannot absorb falls. Lengthen the lens to keep the scene filling the frame, and one power of that loss comes back, exactly. The recovered direction between the cameras does not come back with it. At the same parallax in pixels, a long lens is three times further from the answer, because what fixes a direction is the angle the marks span and no lens can widen that.
The rectangle behind the lensA square closes the line of cameras
A box's three right angles leave a line of cameras, and a square on one of its faces closes it, because the square's diagonals are a fourth right angle the box did not already contain. From exact marks the whole camera comes back. From ordinary marks a 1.33 squeeze is read to two per cent, with a small bias, and a rectangle taken for a square reads as a squeeze of almost its own ratio.
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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 camerasA mirror corridor shows the room once, and the room's lines are a reference by where they are
Between two facing barber's mirrors the corridor of self-images goes on for a dozen round trips, but the shop around it does not: the floor never appears in a level photograph, and the walls appear in the first reflection only, because every later path has to start and end on glass and only what stands between the two panes can be reached. That one sight of the wall behind the sitter is enough. A rail and a door jamb on it, fitted with the self-images, read each mirror's own lean and turn to about a twentieth of a degree with the camera's heading, pitch and roll unknown — better than a camera believed level and a tenth of a degree out. The reference is not the lines' angles, which can be 0.2° off at almost no cost, but their places: a rail 5 mm higher than believed reads as a lean of 0.14°.
Measuring from one pictureA third photograph closes the hole the second leaves, if it brings a building
A second photograph of a row of posts reads every post's lean except the one whose two sight lines run nearest parallel, and that post sits at the end of the row away from the second camera. A third photograph from the other side puts its own worst post at the other end, so between them no post is left: sixty degrees round either side, the worst post is read to 0.10° where either picture alone left it at 0.23 or 0.20. But the third picture does not buy what was hoped of it. Without buildings of their own, three pictures catch fewer of the hidden leaning posts than two at a right angle with one, and the spare equation each post gains cannot read the further pictures' verticals: a lean the whole row shares, or one that grows along it, looks exactly like a vertical misread.
Through water and glassA crooked stick hides in a quarter pixel, and its residual finds it only below a tenth
Eight sticks stand in still water, one of them bowed three millimetres and the rest one, and a refraction fit holds every stick to a straight line. Read exactly, the bad stick carries the largest share of the fit's residual in about seven sets of ten. Read to a quarter pixel, as a careful photograph is, it carries it in one set of five, barely more than chance: a millimetre of bow moves the index more than the reading error does and leaves its rays missing by a twelfth of a quarter pixel. The residual stays blind until the bad stick is bowed five or ten millimetres, and dropping the stick it ranks worst does nothing measurable for the index.
The real instrumentTwo circles leave the eye on a circle
A picture of the ground holds five numbers, and a camera has six. Any number of circles at any distances draw identically from every eye on one circle in the side view, so a second circle cannot give the pitch. It gives the circle. A tape measure cuts it, and which length to tape depends on the pitch.
The second eyeThe road under a car ahead gives its growth a distance, while the road stays flat
One camera that knows its own height reads the row where the car in front meets the road, and that row is the camera's height seen from the car: a baseline of 1.3 metres standing on the road, read in every picture. With the car's growth it ranges a car 20 metres ahead to 0.80 per cent a pixel, six times better than a stereo pair's disparity, and reads the car's width to 0.74. A camera height a few centimetres out scales the distance and leaves the time untouched. A road that starts rising by one degree between the camera and the car moves the distance by up to 24 per cent, and neither horizon puts it right. The lane's own edges bend where the road does and read the rise back.
What a pair is forThe scale names the mistake when the camera has lost its place along the wall
A third camera that knows how far it has paced towards a facade, and nothing about where along or up the wall it stands, still names a stereo pair's column mistake from one picture. Any offset at all refutes the honest reading, which predicts none from anywhere; and once the unknown place is fitted away as one uniform slide, what is left is the rebuilt facade's scale, which each mistake predicts differently. Five centimetres towards the wall names the mistake 95 times in 100 at half a pixel, ten centimetres every time — where a camera that believes it knows its place and is 20 cm out manages 77. The distance has to be paced, though: a rangefinder or the windows' known spacing would have named the mistake by themselves.
Many pictures at onceOne camera known to be level ties the groups; a level phone picture does not read its centre
A phone picture held level puts its centre on its horizon, so it seemed that being level would give back the number an unknown lens takes away. Near square it barely does: the horizon is where the wall's rows of marks meet, thousands of pixels out of the frame, and even an exactly level picture 10° off square reads its centre to ±61 px. But being level does something the question did not ask of it. The wall's twist is the two survey groups pitching against each other, and one survey camera that knows its own pitch against the wall's verticals to a tenth of a degree holds its group still: the twist is reported to ±1.37 mm, as well as a calibrated square-on picture manages. Two such cameras, one in each group, also cancel the false twist a wall out of plumb would otherwise put in.
The eye that movesThe rows read where the camera rides on the body, and a camera behind the centre hides the pitch
A scroll camera's rows over a rough road say how far ahead of the body's centre the camera stands, as well as how the body pitches and bounces. One 800-metre pass reads a camera 1.5 metres ahead to ±0.15 metres, costs the pitch's frequency nothing to do it, and removes the bias a fit that puts the camera at the centre carries — a 1.4 hertz pitch read as 1.54 from 2.5 metres ahead, and as 1.80 from a metre behind a body that does not bounce. Behind the centre the camera's rise works against its turn, and a metre back on a body that does not bounce it very nearly cancels the pitch where it rings.
Systems that kept the measureA walker who stops is ordered by the way it turns
Three copies of a walker who pauses on one spot stand a few centimetres apart, and no reading of where they stand can order them. The way they face can — but not by the rule that followed a walk that keeps moving, which reads a three-copy pause turning 90° only a quarter of the time. A reader that ignores the direction of legs shorter than a step, and charges those legs instead for the turn between the copies' facings, reads it 92 times in 100 and costs a moving walk almost nothing. The middle copy of the pause is the hard one, and it is held only where the walker turned well clear of the error in reading each facing.
What each system gave upScatter breaks the later ties and never the first
Two painters' rates of tiring sign their strips, except where the two creeps meet. If a hand's slips also grow with the strips it has painted, a tied pair differs in its scatter unless both are first strips. Five rows a strip settle a later tie about two times in three, a gap of two strips four times in five, and a run of panels nearly always. Two first strips stay a coin toss, whatever is measured.
What a machine computesA clamp keeps the sky clean and leaves the rolling horizon smeared
Clamping a jittered history to the range of each pixel's neighbours is the renderer's standard guard against a moving edge smearing, and on a rolling horizon it does half the job. Two rows from the horizon, where the neighbours agree, it pulls the smear back at once: a sky row the history had darkened to 0.88 is restored to 0.99. On the rows the horizon crosses, the nine neighbours hold sky and ground together, the window runs from 0.12 to 1.00, and the smear passes through it untouched. The crossing with a plain centred sample moves from 1.5° a second of roll to 1.8; at 16° a second the history is 0.070 from a camera instead of 0.093, still nearly three times a centred sample's 0.025. And a clip tight enough to reach the smear brings the jitter back: on a still camera it pops four times as hard.
The rectangle behind the lensOne pair of frames reads the speed and the bend from the roadside
A camera that reads one frame from the top down and the next from the bottom up turns every roadside pole's rolling-shutter lean round and leaves its own tilt where it was. Half the difference of each pole's two leans is the shutter's alone, and across a row of poles it lies on one straight line in one over the range: its slope is the car's speed and its crossing is the bend. So one pair of frames, a fifteenth of a second, reads both — the speed to 3 per cent and the bend to ±5.6 m from sixteen timber poles, whatever they lean. With the speed known, sixteen poles read the bend to ±2.4 m, a little better than one plumb post followed for sixteen frames. The nearest pole decides most of it.
The second projectionA walker charged at its worst place gains where it looks
A projector sharing one layout between two walkers, drawn for the worse-served of them, took the further walker below the even picture at its worst place — the picture's far edge, which it sees at a slant from five metres. Scored instead at the part of the picture straight in front of each walker, the same layout gives that walker 10 per cent more than the even picture, not 6 per cent less: the samples were taken from where it was not looking. The four rules for sharing then nearly agree, 11 to 12 per cent for the worse-served walker. Two things bring the charge back. A walker looking 30 to 40 degrees ahead along its walk looks at the edge the layout starved. And a walker looking at the picture's middle is served alike by every layout, because every layout the projector can afford takes its samples from there.
Drawn confidentlyThe rule's bow is read by its shape
A straightedge on a painted pavement's diagonal convicts the constant-ratio rule by the size of its corners' worst departure, and gives out once the painter's hand scatters by a third of the bow. The bow has more than a size. It is one smooth arc, to one side, fixed by the ratio, and a hand's scatter is neither. Fit that arc with one free amplitude and the rule is ranked above a construction 96 times in a hundred on five braccia at two pixels, where the straightedge managed 83 — and the reading holds to half the bow's size in scatter.
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Refraction · 12 Habit · 11 Scale · 11 Epipole · 10 Exposure · 10 Homogeneous · 10 Keystone · 10 Outlier · 10
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 from a picture 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.