Visual Plane
every picture states the point it is correct from
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The thread: The viewer is in the geometry — page 5

A perspective picture is a projection from a point, and that point is a fact about the picture — computable from its focal length and the width it is shown at. Every figure states the distance it is correct from, because leaving it out is what makes the subject feel like a matter of taste. Essays 97 to 105 of 105.
the picturehorizonprincipal pointread as levelthe camerasolid: the circle's arc · dashed: the verticals' circlepitched 20° down The real instrument

A circle reads a tilted camera's column, and its tilt has to come from the verticals

The picture of a circle on the ground meets the horizon at c ± i·f for a level camera. For a camera tilted down it meets it at c ± i·f/cos(tilt): the column is still exact, and the imaginary part is the eye's distance from the horizon line rather than from the picture. Every eye on an arc of that radius about the horizon fits the circle — and fits the building's right angle too, which is one of the circle's own. A handheld tilt of five degrees costs a level reading 0.4 per cent; looking 45° down at a manhole cover it costs 41. The building's upright edges, meeting far below, cut the arc once and give the camera back, and from then on the circle is the better half: looking steeply down it names the column to 1.7 px where three vanishing points manage 11.

 25507510002468seconds into the walk, a metre a seconddetail where each walker looks, px a degreenear walker, worse-served rulefar walker, worse-served rulenear walker, smaller-share rulefar walker, smaller-share rulefar walker, even picturescored straight in front of each walker2 m and 5 m from the wall The second projection

A walker charged at its worst place gains where it looks

A projector sharing one layout between two walkers, drawn for the worse-served of them, took the further walker below the even picture at its worst place — the picture's far edge, which it sees at a slant from five metres. Scored instead at the part of the picture straight in front of each walker, the same layout gives that walker 10 per cent more than the even picture, not 6 per cent less: the samples were taken from where it was not looking. The four rules for sharing then nearly agree, 11 to 12 per cent for the worse-served walker. Two things bring the charge back. A walker looking 30 to 40 degrees ahead along its walk looks at the edge the layout starved. And a walker looking at the picture's middle is served alike by every layout, because every layout the projector can afford takes its samples from there.

 no single viewpoint — the rays miss by no distance at all — the rays are parallel4 pictures on 3 kinds of face Where to stand

A flight that has ends

A real staircase has three families of faces and the rule proposed last round predicted three pictures. It supports four, because a picture is fixed by a direction and the flight's two sides point opposite ways — one kind of face that no single direction reaches both of. The rule counts words and the object counts orientations.

 the hingepitched 20°height 1.60 mside view, the ground as a linek = 1.703 m, every eye the same picture The real instrument

Two circles leave the eye on a circle

A picture of the ground holds five numbers, and a camera has six. Any number of circles at any distances draw identically from every eye on one circle in the side view, so a second circle cannot give the pitch. It gives the circle. A tape measure cuts it, and which length to tape depends on the pitch.

 051015200204060how far off the fovea the line passes (degrees)departure from a straight chord, in degrees of visual angleschematic eyeconcentric retinaa line spanning ±65° of azimuth17.30° at 40° eccentricity Surfaces that are not flat

The eye is a picture surface too

A retinal sphere behind an off-centre nodal point takes every measurement this collection puts to a lens or a screen, and answers all of them. It is the equal-area fisheye to within 110 micrometres of retina rather than the equidistant one everybody draws it as, and a flat picture at its own correct distance leaves the identical arc on it, to 1.4e-14 degrees.

 the picture plane, in planto the observer, 208 px further downcorrect from 12 cm, at 160 mm widestation 520 px Constructing a view

Two rules for one pavement

Vignola set out two rules for laying a tiled floor and asserted that they agree. Executed from the same ground line and the same free parameter they agree to 2e-13 px; executed from the numbers their own wordings invite they part by 24 px. The quantity that separates them is the distance the reader has to stand at, and neither rule names it.

 projectorthe seatcorrect from 19 cm, at 160 mm wide23.6° worst Surfaces that are not flat

A projector that is not at the dome's centre

A projector 0.40 of a dome's radius off centre puts its own picture up to 23.6 degrees from where it belongs, and the pre-warp that corrects it is exact for one seat and only one. Two metres from that seat costs 11.5 degrees of the same displacement, wherever the projector itself stands — because the correction never knew where the projector was in the first place.

 horizonthe middle of what is leftthe principal pointcorrect from 22 cm, at 160 mm widecrop 20% · centres 81 px apart Constructing a view

The centre of the picture is not the centre of the paper

A crop translates the image rectangle, so the picture's optical centre leaves the middle of the sheet and the focal length does not move — 81.3 pixels apart at a fifth of the picture, with the horizon at 62.5 per cent of the print. A reader who takes the paper's middle for the picture's stands 2.36 cm out of position, which is 4.9 degrees of the wrong direction.

 a plan, looking down: the object on the left, the frame, the hook on the rightthe hookthe frame, edge-on160 cmplan at 0.25 px per mm · a hook 160 cm behind a 56 cm framecorrect from 46 cm at 160 mm wide Constructing a view

The hook is the centre, and the eye is not

Dürer's string frame projects from a ring of iron driven into a wall, so its centre of projection has a position anybody can measure with a tape. A hook 120 centimetres behind a frame 56 centimetres wide makes a drawing correct from 34.3 centimetres shown 160 millimetres across, with its principal point 161 px off the middle of the sheet. The gridded veil projects from a head instead, and 30 millimetres of head moves its marks by 13.27 millimetres.

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