Concept

least squares — where it appears

4 essays name this object, across 4 fields. What follows is each of them, and the objects they name alongside it.
horizon — the image of the line at infinityvanishing point at x = 1464 — off the framecorrect from 26 cm, at 160 mm wide34° across

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.

foundations · vanishing
the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m

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.

refraction · nocentre
fitted k₁ = -0.280000true -0.280000, off by 5e-15

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.

lens · plumbline
00.50011.50010203040correspondenceepipolar error at every OTHER point, after one match is movedmedian 0.54 pxthe moved oneevery point wrong: median 0.54 px, worst 1.7 pxclean fit 2.3e-13 px

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.

depth · outlier

Named alongside it

The objects these essays reach for when they reach for this one.

ResidualVanishing pointBaselineCamera calibrationCausticcentre of projectionConditioningCorrelationCorrespondenceCross ratioFocal lengthFundamental matrix

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