Cylinder — where it appears
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Area scaleConformalPicture planeAnisotropyArc lengthConeDevelopableEquidistantEquisolidFisheyeGaussian curvatureIsometry