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

Metric structure — where it appears

Knowledge of angles and of ratios of lengths in different directions on a plane, which is everything a Euclidean measurement needs except an overall scale. On a projected plane it amounts to knowing where the two circular points sit on the vanishing line, which is two numbers a picture withholds.

Named by 5 essays across 2 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 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 cross-ratioratio acrossratio recedingright angle45° anglelengtha length 10% wrongexactexactexactexactexact10.0%a repeat's shape 10% wrongexactexactexactexact2.73°exacta repeat 3° out of squareexactexactexact3.00°1.46°exacta focal length 10% wrongexactexactexactexact2.69°0.1%a horizon 5 px outexactexact2.1%0.35°0.35°2.0%ground sloping 3°exactexact14.3%2.30°2.35°2.0%a road crowned 10 cm2.3%1.8%exact0.64°1.59°1.3%one picture, one set of marks, six readingsexact to the arithmetic floor, or not

A wrong model shows at its own level

Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.

metrology · Scale -0.50000.500-10-505the ground's depth test (per cent from one)the wall's test (per cent from one)slope 3°horizon -15 pxhorizon +15 pxa 6 m wall facing the camera; ellipse: 3× the scatter at 0.4 pxtwo mistakes, two lines

A plumb wall tells a slope from a horizon, faintly

A sloping ground and a misplaced horizon fail the ground's depth test alike. A wall standing plumb beside the ground separates them in kind: a slope leaves the wall's test at 1e-16, and a horizon moves it. But it moves it a tenth as far as it moves the ground's — 0.20 per cent for a horizon five pixels out, against 2.11 — so a pair of windows a metre wide cannot see the difference, and a whole facade six metres wide can tell a one-degree slope from its matching horizon 95 times in a hundred. A wall running straight away from the camera sees nothing at all.

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

Named alongside it

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

Affine structureCross-ratioProjective stratificationsingle-view metrologyVanishing lineHomographyMetric upgradeModel errorSimilarityAspect ratioCircular pointsConditioning

All concepts