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

Sphere — where it appears

A surface of constant distance from a point, whose Gaussian curvature is nowhere zero and which therefore cannot be laid flat by any means. That is why no map of it is faithful, and why every choice of chart for a spherical panorama gives something up.

Named by 2 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 the floor, in placeunrolled — 2.611 m of surfacea ridged floor, k = 0.062.6 m of plan is 2.611 m of floor

The floors that unroll

A ridged floor curves visibly and can be laid flat without stretching anything — 7.4e-9 of strain across the patch. A dished floor curves less and cannot be laid flat by any means whatever. The difference is one number, Gaussian curvature, and it is the number Gauss proved no bending can change: 0 for the ridge, 0.0144 per square metre for the dish, and no cleverness in the flattening touches it.

curved · Developable

Named alongside it

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

Area scalePicture planeArc lengthConeConformalCylinderDevelopableEquidistantEquisolidfield of viewFisheyeGaussian curvature

All concepts