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

Gnomonic — where it appears

The projection of a sphere from its centre onto a tangent plane, and the only kind sending great circles to straight lines. It is the flat picture plane under another name, and it is why a photograph of a straight wire is a straight mark whatever the camera is doing.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces

No picture surface keeps everything

A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.

curved · Surface fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples

The arcs a curvilinear drawing uses

The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.

curved · Curvilinear 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 planeevery line—cylinderone parameter · parallelno meeting pointstereographicone parameter · concurrentthey meet to 1e-12equirect.one parameter · parallelno meeting point3 of 4 keep a curvethe signature has three values, not eight

The lines a surface leaves alone

Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.

curved · Straightfamily 42.0°27.3°13.5°13.5°27.3°42.0°leftfrontrightbackupdownmidpoint: 0°7 straight lines, at 60° to the seamworst 42.04° · midpoint 0°

The kink at a seam

A cube map is six flat pictures, so every great circle is drawn exactly straight inside a face — to 3e-15 of its chord — and breaks at the join. The break is a kink and not a bend, it is exactly zero at a seam's midpoint whatever the slant, and it is bounded by 2·atan(½) = 53.130° at the corner.

curved · Cubemap

Named alongside it

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

AnisotropyPicture surfaceArea scaleBeltramiConformalCube mapDemonstrationEquirectangularfield of viewSeamStraight familyEquidistant projection

All concepts