Degenerate configuration — where it appears
Named by 25 essays across 6 fields — each of them below, with the objects they name alongside it.
A flat scene fixes no second eye
Eight marks on one plane leave the eight-point design matrix two ranks short, so a two-parameter family of fundamental matrices satisfies every mark exactly — three of its members are 1.41 apart after normalisation and all three fit to 1.6 × 10⁻⁶ pixels. The same photographs determine the plane's own map from four marks, to 5.9 × 10⁻¹³ pixels.
How flat is flat enough
With exact marks the transition has no width at all — 84° of pose error at exactly coplanar and 0.000° at eight parts in ten thousand of relief. Put three tenths of a pixel of reading error in and the same sweep becomes a slope three decades wide, crossing into usefulness when the out-of-plane parallax reaches about ten times the marking error.
Square to the camera is the worst mirror
A mirror pair's baseline runs along the mirror's normal, so a mirror facing the camera puts the second eye directly behind the first — the forward-motion arrangement, with the epipole in the middle of the frame and the rays to a mark crossing at 23°. Turning it forty-four degrees opens that to 65° and cuts the worst depth error threefold, and the number to watch is not the angle but where the reflected lens sits on the print.
Two marks off a known plane find the other eye
Map a courtyard's ground from one picture into the other, and every raised mark lands somewhere the map did not send it — displaced along a line through the image of the other camera, to a fifth of a billionth of a pixel. Two such marks put that image where it is, and with it the whole epipolar geometry.
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.
A third eye that lands on the next post
Match one post of a railing to its neighbour and the pair reports it at 19.8 metres instead of 9.0, with every test two photographs can run at the arithmetic floor. A third picture usually exposes that by hundreds of pixels — but at five azimuths in seventy-eight degrees the wrong point lands within three pixels of another post, and the third view confirms the mistake. Narrow the railing to twenty centimetres and those places cover 28 per cent of the arc.
A parallax length is a height over a depth
After a known plane's map, every raised mark's displacement points at the other camera's image, and its length carries the mark's height above the plane over its depth — but not as the ratio of lengths it looks like. That ratio departs from the point's own number by up to 45 per cent. Read as a coefficient on the epipole, the same length gives height over depth from the first camera to four parts in a hundred trillion, the same from every second picture.
A railing matched one along stands underground
Match every post of a railing to the next one along and two photographs build a perfect railing at 19.8 metres instead of 9.0 — straight, evenly spaced, every post on both its rays. Nothing near it can say so. What convicts it is what it stands on: its feet are 1.32 metres under the ground, and one ground mark read to a pixel shows every foot a whole period out from sixty metres away. Its length helps only through its ends, and once those leave the frame, counting matches prefers the wrong railing.
A turn moves the picture, not the blind point
A vehicle that yaws for a moment between frames points its camera somewhere new without taking it anywhere new, and that is exactly what a turn cannot help with. The mark it is driving toward stays blind at every turn; the disc around it moves by two hundredths of a pixel for a 3° yaw; with a sway, a turn changes a mark's depth cost by 0.2 per cent. Aiming the camera off the road moves the blind point across the picture — and off it past 30° — but the thing it blinds is still the thing the vehicle is heading for.
A facade matched a column out comes forward, not back
A railing matched one post along is rebuilt 2.2 times further away. A facade of windows 2.4 metres apart cannot be: a one-column shift that way needs a negative disparity, and the only mistake left in front of the cameras brings the facade to 6.3 metres from twenty, a third its size, hanging 1.1 metres off the ground with nothing under it to sink into. Its corners hold it only if the wall is assumed flat. Two corners then convict it at every period; one corner and a drainpipe need the pipe half a metre in for windows and four and a half for a fine repeat.
Steering swings a camera too little to see ahead
A vehicle yaws about its rear axle, so a camera mounted ahead of the axle is swung sideways whenever the vehicle steers — a real baseline, where a turn alone gave the blind point ahead nothing. Half a degree of yaw swings a windscreen camera 1.3 centimetres and a bumper camera 3.1. Straight ahead at eight metres a pixel then costs 96 and 40 per cent of the depth. The cost falls as one over the arm times the yaw, and a usable 10 per cent needs 7.2 metre-degrees: two degrees of steering every frame on a bumper. Ordinary lane-keeping gives a fraction of that.
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 raised eye reads the storeys as the pair read the columns
Put a third camera 3.5 metres above the left one and its epipolar lines run up the facade: the storeys repeat along them as the columns repeated along the level pair's rows, the roofline and the foot carry depth where the corners did, and a raise shorter than a storey can only pull the windows nearer. The three pictures together refuse the level pair's one-column mistake at every raise except those where the storey over the raise equals the period over the baseline — 1.375 and 2.75 metres here — and even there the facade's lowest storey lands on the ground.
A drift across the lane closes the blind point, slowly
A car that wanders twenty centimetres either side of its lane every four seconds moves its camera twenty-eight centimetres sideways in a second, far more than steering ever swings it. Looking back over that second, the road twenty metres ahead is known to 23 per cent a pixel; over four seconds, to 10.5. The error falls as one over the wander and grows with its period and the car's speed, so a car that keeps its lane well, drifts slowly or drives fast leaves the point it is heading for nearly as blind as before.
What 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.
A stereo pair and a lane wander add as two readings of one depth
A pair of cameras 0.3 metres apart across a car ranges the road twenty metres ahead to 15.8 per cent a pixel from one moment's two pictures — and to 4.6 per cent from the last second's sixty, because every picture is another reading. Add the car's ordinary drift across its lane and the second's pictures give 4.5: exactly what the pair's look and a single camera's wander would give as two independent readings. The two baselines neither waste each other nor multiply, while the pair's is the larger; only a narrow pair on a car that drifts half a metre does better than the rule, by a quarter.
A climbing eye names the column mistake, if it knows how high it is
A level pair that matches a facade a column out rebuilds it nearer, and a third camera climbing above the left one watches the rebuilt windows slide down its picture at 98 pixels a metre for every column of the mistake. Read against the real windows, that slide is a ruler: a metre of climb from level names a one-, two-, three- or four-column mistake every time. What it costs is not frames but knowledge — the climb must begin before the rebuilt facade slides off the lattice, and the camera must know its own height to about a quarter storey over the mistake's slide, 34 centimetres for one column and 9 for four.
A staircase winds the mistake round the lattice
A third camera climbing a stair along a facade sees a stereo pair's column mistake slide through its picture along the stair's own direction, wrapping at the columns across and the storeys up — a line wound round a torus. Such a line comes back onto a real window only when the stair's rise over its run is a ratio of whole storeys to whole columns; a degree off the diagonal, no two readings come within 3 px. The stair names the mistake from one frame, as a vertical climb does, but its readings sit twice as far apart in a frame, so a camera that knows where it is only to 10 cm still names it 95 times in 100 against 82, and a 30° stair keeps naming it a metre and a half along, where a vertical climb has lost half.
A car ahead costs a pair its speed, and a car that may drift costs a single camera everything
A stereo pair's last second of pictures ranged the still road twenty metres ahead to 4.5 per cent a pixel. A car twenty metres ahead is not still, and each old picture saw it somewhere else. Read together with its speed, a car keeping pace is ranged to 4.5 per cent all the same — it stood twenty metres from every picture, and reading its speed spends exactly what that was worth. A car that may also drift across the lane costs the pair a fifth more and takes a single wandering camera from 38 per cent to 185. A braking car caps the useful look at about a second.
A turn at the landing makes the mistake a scale, and a scale is weak evidence
On a straight stair along a facade, a stereo pair's column mistake slides through a third camera's picture as one offset shared by every window. Turn the stair towards the wall at a landing and the offset stops being shared: the rebuilt facade's picture becomes the real one's scaled about the point the camera is walking towards, a different scale for every mistake, so no picture on that flight is blind. But the scale grows as the walk over the rebuilt facade's distance, and 10 cm towards the wall separates the one-column reading from the honest one by 1.8 px where 10 cm along the stair gives 9.8. Turning the camera in place adds nothing at all. What the turn does buy is tolerance: an error in how far towards the wall the camera has come costs almost nothing, for the same reason.
A growing car times the gap without its width
A pair of cameras ranges the car in front from its disparity, and reads how soon the gap will close only as well as it reads the car's speed: to 39 per cent a pixel from a second's pictures, for a car 20 metres ahead closing at 5 metres a second. One camera watching the same car grow reads the same four seconds to 8.8 per cent, without knowing how wide the car is. The car's own width is a baseline six times the pair's. The distance needs the width, and the pair, reading both, measures the width as well.
The 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.
The arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
A dent breaks the terminator
A ball's lit boundary is one closed curve, 313.2 cm around; press a dimple into its top and, at 45° of elevation, the walk that traces it finds 181 crossings where a point still faces the light but is occluded by the body's own rim — a second curve, 23.1 cm long, that a convex surface can never produce. The two curves meet where the dent's own deepest point goes dark, at 50.952°, with the ray's clearance reaching zero at a stationary point rather than merely a small one.
A shadow edge read as a profile
A lamp, a stick and a camera recover a stepped object's profile to 4.9e-15 m rms when the marks are exact, and to 10.4 mm once they are read to two tenths of a pixel — the same linear law a fitted exponent of 1.001 confirms. What actually sets that number is the angle between the sweeping light plane and the camera's own ray — the amplification is least, 17.0 times a pixel, broadside at 6°, and grows without bound toward -36.1°, where the plane contains the camera's own eye and the recovery keeps none of its marks at all.
Named alongside it
The objects these essays reach for when they reach for this one.
TriangulationBaselineEpipoleCorrespondenceOutlierrobust estimationDepth uncertaintyEpipolar lineMoving viewpointConditioningEpipolar geometryHomography