Visual Plane
every picture states the point it is correct from
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The thread: One picture, one length — page 3

A photograph gives up every ratio in the scene and no size at all: scale the world and the eye together and not a pixel moves. So a measurement from a single view is a ratio until the reader supplies one length, and which length that is decides how good the answer can be. Essays 49 to 56 of 56.
plan: the apex has movedfree along the kernelisometricrecovered to 1e-16 The other systems

The drawing that gives the solid back

One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.

 vanishing point 1vanishing point 2where the camera wasassumed centre 50% alongfocal 1129.9 px · 0.283 : 1 Constructing a view

The arc every eye stands on

Four drawn corners known to be a rectangle fix the horizon of their plane and nothing else. The eye that drew them has to see the two vanishing points at a right angle, so it lies on the circle those points are a diameter of — and every point of that arc reconstructs a genuine rectangle, with right angles to five parts in ten million million of a degree, and a different proportion.

 00.2500.5000.750120406080assumed centre of the picture, % of the way between the vanishing pointsreconstructed proportion of the rectanglethe rectangle that was there, 0.667 : 115.7× across the arcevery one a true rectangle Constructing a view

The proportion is the assumption

Read the proportions of a rectangle out of a photograph of it and the answer is a function of where the centre of the picture is assumed to be. Sweeping that assumption across the horizon takes one drawn quadrilateral from one part in fourteen to slightly wider than square, every reconstruction a genuine rectangle, and only a fiftieth of the sweep within five per cent of the truth.

 00.2500.5000.75010.2500.5000.75011.25the largest stretch a painter will accept, log₁₀share of the design that is inside ita flat floora cornera cluster of blocksa seating rakean ascending flighteye at 1.70 m, 2.4 m in front of the picturethe cap a painter will accept Where to stand

The stretch decides the band

A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.

 -3-2-1010123how many measurements are averaged, log₁₀the error in the recovered height, log₁₀ mmthe spread — -0.52the bias — flat, and microns1 px on the clicked topspread 0.177 mm at m = 1024, bias 20.0 µm What survives

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.

 the horizoncorrect from 16 cm, at 160 mm widecross-ratio 1.366025 Constructing a view

The stair that turns has a vanishing point that moves

A spiral stair's treads are one rectangle turned by a constant angle, so every front edge has its own vanishing point and the twelve lie on one horizon to 2.8e-14 pixels. Four consecutive of them read a cross-ratio of 1.353720, and the formula behind it holds no focal length, no principal point and no eye — so a photograph gives the builder's 12° turn back.

 18.0 cmcard at 1.00 m144.0 cmcard at 2.40 mrecovered: 36.0 cm at 3.00 mtwo cards, two bands, one lampcondition number 37.1 Light and mirrors

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 heads' linethe horizoncorrect from 16 cm, at 160 mm wide59 px, 1.46 m at the far figure Constructing a view

Figures on a street that slopes

Equal-height figures have their heads on one line, and the taught rule says the line is the horizon. On a street rising at 8.33 per cent the heads are still collinear to 2.8e-14 pixels and the line is 58.9 pixels above the horizon — the street plane's own vanishing line. The taught rule loses 1.46 m of a 1.62 m figure at the far figure, and runs out of figure altogether at 19.4 m.

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