Visual Plane
every picture states the point it is correct from
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The thread: The surface is a choice

A picture has to be cast onto something, and the something is decided rather than given. Flat keeps straight lines straight and cannot reach 180°; a cylinder reaches all the way round and bows every horizontal; stereographic keeps every angle and no scale. The price of each is a number, and the corner where a surface pays nothing is empty.
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 Surfaces that are not flat

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.

 fitted conic: 1.467 from being a circlecylinderthe samples, fitted Surfaces that are not flat

The cylinder, and the price of going all the way round

A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices.

 fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples Surfaces that are not flat

Stereographic keeps every angle, and only stereographic

One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at.

 02468020406080angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame Surfaces that are not flat

Every fisheye is a different rule

The word "fisheye" names a shape of lens and not a projection. There are several, they disagree with each other by tens of per cent at the frame edge, and each is the right answer to a different question — one is a protractor, one is a counting instrument, one preserves shape. Which one a lens implements is a fact about that lens, and it is rarely printed on the barrel.

 no picture at 300°the plane is unbounded at 180°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%300° across in every panelsame scene, same angle, six surfaces Surfaces that are not flat

What a 360-degree photograph actually is

The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it.

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