least squares — where it appears
Named by 31 essays across 12 fields — each of them below, with the objects they name alongside it.
Where parallel lines meet
They meet at a point that is not in the plane, and the horizon is the line made of all such points. Treating that as bookkeeping misses what it buys — a vanishing point becomes an ordinary point of the picture, findable from the drawn lines, checkable, and enterable into any calculation the others are.
A picture through water has no viewpoint
Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.
Fitting a lens from straightness alone
No calibration target, no known scene, no camera. Only the knowledge that some edges in the picture were straight — and the coefficient comes back to fifteen digits. Then it comes back with a companion, and the two are correlated at −0.997.
The centre a scroll does not have
Fit a common point to the rays of one section of a handscroll and it misses by metres. The miss is not a residual to be tightened — it is exactly the standard deviation of the eye's own track, it grows linearly with how much is unrolled, and it goes to zero only for a section of no width.
A wrong match is not a small error
Move one correspondence of forty-four by thirty pixels and the recovered geometry is wrong for every other point — the typical one by half a pixel, from a fit that was exact to a part in ten trillion. Least squares has nowhere to put a bad row except across all of them.
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.
The curvature a shadow reports
A flat floor makes a shadow a homology, so four marks predict the rest exactly and a curved floor mispredicts them by millimetres. Turn that round and the mispredict is a measurement of the floor. It recovers a dished floor's own curvature to a per cent — and returns 0.482 for a floor made of two planes, whose curvature is zero everywhere, with a residual of ten microns.
A pane gives a product before it gives two numbers
A flat pane of glass displaces every point it is seen through, and the displacement at small angles is the thickness times one minus the reciprocal of the index. So the two numbers arrive multiplied together. Four panes from 6.5 to 13.2 millimetres thick, with indices from 1.35 to 2.1, agree to under two microns over an eight-degree fan and separate by more than a millimetre over sixty — and a fit over the narrow fan returns whichever pair it started near.
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.
A fitted radius is wrong before it is uncertain
A sphere and a paraboloid of the same vertex radius agree to second order, so a fit over a small aperture cannot separate them. What it does instead is return a confident radius that is wrong by a stated percentage, with a residual far below any measurement floor — 0.03% of bias behind a residual of three ten-thousandths of a degree. The residual only clears a two-hundredth of a degree at six times the aperture, by which point the bias is thirty-six times larger.
A pond in a scroll is not an ellipse
A pinhole draws a round pond as an exact ellipse whose widest row is 3.58 px off the row of the pond's centre — the drawn-circle error every perspective textbook warns about. A handscroll draws the same pond widest exactly on its centre's row, and draws it as a quartic that no conic fits: the best ellipse misses it by 3.08 px. Each keeps what the other loses, and off to one side the pinhole's pond leans 9.67 px while the scroll's does not lean at all.
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 lines that calibrate a lens
One straight edge through the centre of a picture says nothing about a lens's distortion, and one 180 px from the centre determines k₁ to 9.0 × 10⁻⁴ — the precision rises in proportion to the offset. But distance from the centre is not enough. Crowd three edges on one side and, the moment the distortion centre is also unknown, the coefficient is ten times worse, because a bend on one side looks like a moved centre; put one edge across the centre and it barely changes.
Undoing a picture made on a curve
Three rounds of work here have measured what a curved receiving surface costs going forward — 5.67 mm on a dish, 9.07 mm on a ridge, 529.4 mm on a vault. None of them asked whether the design can be got back. It can, exactly, and the price is stated precisely: you have to supply the surface. Told the floor, the recovery returns a design to 1.1e-12 mm; told nothing, and fitting the four marks every rectification tool fits, it is 111 mm out.
A mismatch on its own line needs a third eye
Slide one mark of a correspondence 30 px along the epipolar line the other mark fixes, and every test two photographs can run stays at the arithmetic floor — epipolar distance 2.2e-14 px, the two rays meeting to 1.5e-15 m, reprojection 1.1e-13 px — while the point is reported half a metre too near. A third picture exposes it by 21.4 px from a third eye two metres off the first line of sight, and by exactly nothing from an eye on that line.
How well the floor has to be known
“Supply the surface” is an instruction with no number in it, and an instruction with no number is a wish. Here is the number: the error in the recovered design is very nearly proportional to the error in the assumed curvature — the constant varies by 1.4% across a twentyfold range — so about nine parts in a thousand of the curvature buys one millimetre on a design 1800 mm wide, and 40% out costs 43.5 mm.
A third ray is worth what its picture is worth
Three eyes on one point, two at seven metres and one walked back to seventy. The point nearest all three rays in metres is 132 millimetres from the truth and the point of least reprojection error is 34 — the same 34 the near pair gives alone — and the first is pulled 12 millimetres along the line to the distant eye. And arrangement beats count outright — two rays spread over fifty-five degrees beat eight rays inside four, by a factor of 4.4.
The response is at the ends and the information is not
A radial map bows a straight edge by an amount that grows as the square of the distance along it, so 93 per cent of an edge's response to the coefficient lies in its outer quarters. Spending the marks there is 16 per cent worse than spreading them evenly, because two clusters say nothing a shifted, tilted line could not say. What identifies the coefficient is a curvature, which needs three places — both ends and the middle, which beats an even spread by 11 per cent.
The wedge recovered with the camera
Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.
The 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.
A fit weighted by the miss trusts only the surface
Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.
The height a flat floor cannot give
The marks of a floor anamorph name the eye's position on the floor exactly and say nothing about how high it was — every candidate height explains them perfectly, to one part in a thousand trillion. That is a fact about planes rather than about anamorphs. Ripple the floor by six centimetres and the family collapses: the true height explains the marks exactly and the nearest wrong one, five centimetres away, leaves two millimetres on a design 1.8 metres wide.
A wedge moves the centre, not the lens
A wedge of glass in front of a lens deflects every ray by a little more the further off the axis it goes, which looks like the shape a radial distortion coefficient describes. Fitted together, the two are nearly independent — correlated at 0.16 at most — and a calibration that knows nothing of the glass does not invent a lens: it reports a coefficient of about 0.002, moves its principal point by 4.5 to 14 pixels, and leaves a swirling residual that no radial model takes. The blame goes to the camera's centre, and the residual says so from twenty-two degrees off the axis.
Five marks and the sixth
Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.
The residual has a shape
A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.
An error with two terms
Two results from machineries with nothing in common have now found the same shape. A panorama's parallax separates into a term that halves every time the frame count doubles and a term with no frame count in it at all; a silhouette's error into an excess that falls as one over the square of the view count and the area of a concavity that is the same number at four views and at a hundred and twenty-eight. Fitting both terms turns the distinction into a measurement, and pointed at seven of this collection's own laws it reads every one of them the way its own essay does.
The bias out of reach
A single-view height recovered from the cross-ratio has an error with two terms. The spread of an average of m readings falls as one over the root of m, by a fitted exponent of −0.52 over three decades. The bias does not fall at all — and it is eight microns, against a single-shot spread of six millimetres, so the two cross at 535,000 measurements. There is a floor here and no experiment anybody will run can see it.
A floor with a referent
Recover a focal length from two vanishing points and measure more points along each line. Through a pinhole the error falls from 0.34 per cent to 0.05 and the instrument finds no floor at all. Through a lens of k₁ = −0.05 it falls, turns, and rises to 0.70 per cent — because the noise the extra points removed had been partly masking the lens's bend. The floor is 0.72 per cent of the focal length, and doubling the distortion coefficient doubles it to 1.44. It is not noise and not conditioning; it is the model, priced.
Five tangents name the same conic
Fitting a conic to five lines is the same six-coefficient nullspace problem as fitting one to five points, with the roles exchanged, and it lands on the same curve to 2.6 × 10⁻¹¹ pixels. What duality does not promise is that the two are equally good evidence — and measured at equal marking precision they are, to within seventeen per cent, because a drawn tangent is made of points.
The lamp's size over its distance, and nothing else
One straight-edge shadow gives a lamp's angular size and nothing about its actual size or distance — scale a 36 cm lamp and its 3.0 m distance together and the band it casts differs by 0.0e+0. A second card at a different height breaks the tie, recovering 36.00 cm at 3.000 m from bands of 18.0 cm and 144.0 cm alone, at a condition number of 37.1.
The lamp a low shadow cannot locate
Five posts, their shadows, and the line from each foot through its own shadow's tip meet at the lamp standing over them — at 52 degrees up that meet moves 0.11 m for half a pixel of marking error; at 11.5 degrees, 0.64 m. The two eigenvalues of the same pencil of lines part by a factor of 8849084 across the sweep, and half a pixel becomes more than a metre of lamp below 14.4 degrees.
Named alongside it
The objects these essays reach for when they reach for this one.
ResidualConditioningDemonstrationinstrument limitModel errorerror propagationVanishing pointBaselineCamera calibrationDegeneracyGaussian curvatureReconstruction