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

Scale — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. a room 2.8 m across, eye 1.6 m up1 mthe same plan, 137× bigger137 midenticalpicturesthe picture — both scenes, drawn twice, one on top of the otherlargest disagreement 1e-13 px over 8 verticesone length has to come from outside the picture

    The one thing a single view cannot give

    Make the world a hundred and thirty-seven times larger and move the eye a hundred and thirty-seven times further away, and the picture does not change by a measurable amount. Every ratio in a scene is recoverable from one photograph and no size is, and that is not a caveat about the method — it is the shape of the method.

    part 1 · metrology
  2. as recovered0.308 across× 3.4 — same two pictures1.046 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m

    Two views give shape and no size

    Every pairwise distance ratio in a courtyard recovered from two photographs matches the real one to fourteen digits. The courtyard's actual size is not merely uncertain — it is absent, and a reconstruction three and a half times larger fits the same two pictures exactly as well.

    part 2 · twoviews
  3. a length on the ground3.22%1.00 m across the referencethe camera's height0.36%1.62 m above the grounda repeated object, size unknown3.60%the answer in units of the repeata standing object of known height0.41%1.75 m, upright, anywhere on the …the focal length and the horizon1.94%700 px, and where the ground's li…the spread of the answer, per closureshorter is better

    Five facts that close the same gap

    The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.

    part 3 · metrology
  4. 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.

    part 4 · metrology

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