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

Rectilinear projection — where it appears

The flat picture plane as a surface: gnomonic, straight for every line, and with a half-width that runs away as the field approaches half a turn. It is the only surface that keeps every straight line straight, and Beltrami's theorem says no other can.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

54 px69 px84° across27% wider at the edge

Wide angle is not distortion

A wide lens stretches shapes at the edge of the frame by exactly 1/cos θ — 3% at 28° across, 41% at 90°. Every bit of that is what a correct rectilinear projection must do, and every bit of it disappears if the picture is viewed from the point it was made for. Nobody views it from there.

viewing · Widefield d = 0d = 0.5d = 1d = 2d = 4d = 0straight48.06° angle×17.80 aread = 0.54.81% bend23.08° angle×4.66 aread = 17.21% bend16.84° angle×2.65 aread = 29.61% bend23.13° angle×1.57 aread = 411.54% bend32.71° angle×1.73 areaa general straight line · worst angle · area range, over the field130° acrossangular minimum at d = 1.00

One parameter between two surfaces

Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.

curved · Panini worst 1e-13 px outcorrect from 17 cm, at 160 mm wide3 × 3 tiles

A tile is an off-centre frustum

Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.

pipeline · Tiling leftfrontrightbackupdownacross the left/front seam: 1.80°, with each side straight to 7e-16corner area ×5.196anisotropy √3 = 1.7321 there

Six flat pictures of everything

There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.

curved · Cubemap

Named alongside it

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

field of viewAnisotropyArea scaleDemonstrationPicture surfaceStraight familyAnamorphosisBarrel distortionBeltramiCamera matrixCamera tiltConditioning

All concepts