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

Solid angle — where it appears

The area a set of directions occupies on the unit sphere, and the quantity a count made in a picture is usually meant to be measuring. A count made in a picture is a count per unit picture area, so it has to be divided by the surface's area scale before it counts anything real.

Named by 4 essays across 3 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 machinery has been returning it since the day it was written 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 88.9° between the two circlescorrect from 23 cm, at 160 mm widelamp 2.6 m · shadow circle at 80.0°

The edge of a shadow is drawn on the object

The outline of a cast shadow is the image of a curve, and the curve is on the caster. It is not painted there: it slides when the lamp moves, it is not the outline the camera sees, and the two coincide only in the arrangement where no shadow is visible at all.

light · silhouette

Named alongside it

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

Area scaleEquidistantEquisolidFisheyeStereographicJacobiannecessary, not sufficientPicture planeQuadratureAnisotropyConformalConic

All concepts