Visual Plane
every picture states the point it is correct from
Concept

Affine structure — where it appears

The part of a plane's geometry a picture fixes once its vanishing line is known — parallels, midpoints and ratios of lengths along a direction. It sits between the projective structure, which needs nothing recovered, and the metric one, which needs the circular points as well.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

010200.8000.90011.101.20the assumed aspect, as a multiple of the true onethe error — degrees for the angle, per cent for the lengththe angle, in degreesa length into the pictureone wrong assumption, three quantitiesshape and size are different facts

An angle on the ground

An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.

metrology · Rectify 1correct from 19 cm, at 160 mm wide46° across

Seven is not a power of two

Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.

construction · Projectivescale a cross-ratioa ratio along a linean anglea length in metressuppliesa length on the groundexactexactexactexactthe camera's heightexactexactexactexacta standing object of known heightexactexactexactexacta repeated object, size unknownexactexactexact10%the focal length with the horizonexactexactexact10%one picture, one set of marks, four readingsthe free scale set 10% wrong

Where 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.

metrology · Scale 0204060204060how oblique the view of the plane is (°)what one pixel costs, against its cost at 20°a ratio of lengths — affinean angle — metric1 px on four corners, one homographya cross-ratio moves by 0e+0

The ladder of assumptions is a ladder of conditioning

Push the four corners of a board by one pixel and read three quantities through the one recovered map. A cross-ratio does not move at all — it is read in the picture and never went through the map. A ratio of parallel lengths moves by a tenth of a per cent at twenty degrees of obliquity and by 1.6 per cent at seventy-eight. An angle moves by sixteen thousandths of a degree and by nine tenths. The stratification ladder is usually taught as a hierarchy of what is assumed; it is also a hierarchy of what a pixel costs.

foundations · Stratification 00the third mark, at infinity0 … 12 marks in the unit intervalcorrect from 19 cm, at 160 mm widerank 1 · 3e-15 from the exact rationals

What a straightedge reaches on a receding line

Three marks on a receding line fix a coordinate, and everything a straightedge builds from them is a ratio of whole numbers. Three complete quadrangles put 1655 marks on the unit interval with no gap above 0.0038, a third costs two quadrangles where repeated halving never reaches one at all, and the same net built on a line that does not recede arrives at the identical 30 coordinates.

construction · Projectivescale

Named alongside it

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

Cross-ratioMetric structureComplete quadrangleConditioningDyadic rationalHarmonic conjugateHomographyIncidenceProjective stratificationSimilaritysingle-view metrologyStraightedge construction

All concepts