Equidistant — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Also named here as equisolid, solid angle, stereographic — the same set of essays touches all of them, so they are one junction rather than several.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Area scaleEquisolidFisheyeSolid angleStereographicJacobiannecessary, not sufficientPicture planeQuadratureAnisotropyConformalCylinder