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

Crop factor — where it appears

The ratio of the 35 mm frame's diagonal to a smaller sensor's, used to convert focal lengths and exact only on the diagonal. It converts focal lengths between formats and is exact only along the diagonal, because two formats of different shape cannot be matched in both directions at once.

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

full frame · 39.6°APS-C · 26.6°Micro Four Thirds · 19.6°1 inch · 15.0°phone (1/1.7″) · 8.7°one 50 mm lens · the angle is a property of the rectangle behind itMicro Four Thirds: 2.00× diagonal, 2.08× wide, 1.85× tall50 mm across five formats39.6° down to 8.7°

A focal length is not an angle

Fifty millimetres means nothing until a rectangle is named behind it. The same lens is 39.6° across full frame, 26.6° across APS-C and 8.7° across a phone sensor — and the distance the resulting print is correct from depends on the ratio of the two, so two cameras matched on angle agree exactly whatever their formats.

sensor · Format 3 m, 50 mm1.50 m, 25 mmsubject ×1.000000 · background ×0.526 · zoom alone would give ×1 for bothnear-to-far ratio 10.00 → 19.00changing the focal length leaves it at 1.000000000000

Stepping closer is not zooming

Changing the focal length leaves the ratio between any two things in a picture exactly alone — to twelve decimal places, at every focal length there is. Moving changes it. Hold the subject's drawn size across a step from 3 m to 1.5 m and the background halves, which is the whole of the shot everybody knows and nobody derives.

sensor · Dollyzoom worst 1e-13 px outcorrect from 17 cm, at 160 mm wide3 × 3 tiles

A tile is an off-centre frustum

Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.

pipeline · Tiling 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

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.

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

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.

construction · Crop

Named alongside it

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

Station pointDemonstrationFocal lengthPrincipal pointViewing distanceConditioningfield of viewPicture planeAspect ratioCamera calibrationCamera matrixCamera tilt

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