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

Quadrature — where it appears

Numerical integration by summing sampled values against their weights, where the weight for a picture is the reciprocal of its area scale. It is how a figure turns a sampled picture into a number about the world, and getting the weight wrong is how an equal-area test can be passed by a surface that is not.

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

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

Named alongside it

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

Area scaleEquidistantEquisolidFisheyeJacobiannecessary, not sufficientSolid angleStereographicAnisotropyConformalCylinderinstrument limit

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