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

Equisolid — where it appears

The fisheye rule with picture radius twice the sine of half the angle off axis, the one that gives equal solid angles equal areas. It is the rule that makes a count of pixels a count of sky, which is why it is the one chosen when a fraction of the hemisphere is being measured.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

00.2500.5000.7501050100150angle of the direction from the camera's own axis (degrees)where it lands in the picture, as a fraction of the picture's radiusequisolidequidistantorthographica ball 24 cm across, camera 24 radii offequal-area within 0.82% · equidistant 21.5%

A mirror ball is an equal-area fisheye

Photograph a mirror ball from far enough away and its rule is ρ = R·sin(θ/2), which is the equal-area fisheye — not an approximation to it, the rule. Measured, the departure falls from 4.27% of the picture's radius at 3 radii to 0.01% at 2000, while the next-best named rule stays 21% out at every distance. And the ball reflects 100.0% of the directions there are, which no designed surface does.

mirrors · Mirrorball 012020406080angle off the axis (degrees)area printed per solid angle, against its value on axis (log₁₀)equal-areaequidistantcylinderstereographicplaneswept at 45° to the axes, out to 80°equal-area 1.000 · flat plane 191×

The third column is area

This field has measured what each picture surface does to straight lines and to shape. Both are questions for somebody looking at the picture. Somebody counting in it wants a third column, and the same projections have been returning it all along without anybody asking: the equal-area fisheye holds a square degree at one printed area to 8e-8 across 80° off axis, while a flat plane inflates it 191-fold.

curved · Equalarea share of the 70° field counted as cloud — the truth is 10.31%equal-area fisheye10.31% (-0.0%)equidistant fisheye9.37% (-9.1%)stereographic7.56% (-26.6%)flat plane2.40% (-76.8%)401² of picture, four caps of cloud0.01% on the equal-area rule, -77% on the flat plane

Counting cloud by counting pixels

A sky camera looks up and something counts the white pixels. On an equal-area fisheye that answer is right to 0.01%, which is the grid's own error. On an equidistant one it is 9% low, on stereographic 27% low, and on an ordinary flat lens 77% low — against a cover that is known exactly, because the clouds here are caps whose solid angles add. Weighting each pixel by the surface's area scale repairs every one of them to better than a fifth of a per cent.

curved · Equalarea 00.2500.5000.7501020406080angle off the optical axis (degrees)image radius, against the radius the same lens gives at 90°equidistantequal-areastereographicorthographicflat planeeach curve divided by its own radius at 90°54.0% apart at 45°

Which rule a fisheye obeys, from straightness alone

Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.

curved · Fitlaw 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

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.

curved · Retina

Named alongside it

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

FisheyeArea scaleEquidistantSolid angleStereographicAnisotropyConformalEquidistant projectionfield of viewJacobiannecessary, not sufficientPicture plane

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