Field

The rectangle behind the lens

A focal length is not an angle until a piece of silicon of a stated width is named. And that silicon is a grid of samples that need not be square, read over an interval rather than at an instant, mostly a row at a time — so a frame is neither one place nor one moment, and the site's own round trip cannot tell.
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.

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.

0.0 ms8.3 ms16.7 ms25.0 ms33.3 mslean 1.052° drawn against 1.055° predicted · a still world leans 0.000°no single viewpoint — the rays miss by 33.3 ms of travelthe frame is a stack of projections indexed by row

Every row is a different camera

A shutter that reads its rows one after another images each of them from wherever the camera was at that instant, so a frame is a stack of projections indexed by height — a handscroll with the roll running down the picture. Its rays miss their own best centre by the spread of the eye's track, at a ratio of 0.988, and a global shutter's meet to 2 × 10⁻¹⁶ m.

3 m · 18 px6 m · 13 px12 m · 8 pxnear ÷ far = 2.123 against a depth ratio of 2.123exposure 33.3 ms · 6 m/s across the frameeach streak is straight; the set of them is not one kernel

A frame is an interval

An exposure is not an instant, so a frame is an integral of projections and every moving point draws a streak. The streak is straight, because the image of a straight path is straight — and its length goes as one over the depth, so two objects at 3 m and 6 m blur by lengths in the ratio 2.000. No single kernel describes the frame.

pixels 2.00 : 1recovered focal length219.9 pxthe camera's actual one436.4 pxwhat the recovery returnsspread across three estimates 0.0e+0worst bundle residual 9.1e-13 pxboth are what a wrong picture would trippixel aspect 2.00 unmodelledfocal length 49.6% short, every diagnostic clean

The pixel that is not square

A camera with two focal lengths is a real thing — anamorphic cinema optics, non-square photosites, a stretched video format. Hand this site's own round trip a picture from one and it returns a focal length 49.6% out, a principal point far from the truth, three independent estimates agreeing to 1e-16, and bundle residuals at the noise floor. Every alarm the site has stays silent.

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