Area scale — where it appears
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.
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.
The sky inside a cone
From under water the whole sky — every direction out to the horizon — arrives inside a cone of 48.61°. Outside it the surface is a mirror. That cone is a picture surface, and it has a distortion no surface in the curved field has — an area scale that runs to zero.
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.
The cylindrical mirror unbends it
Put a mirrored cylinder in the middle of the sheet and the light path from the eye to the paper bends once. The map that results is not a homography and not a projection in the plane sense at all — it varies its scale by a factor of seven across the design, which is why the smear is unreadable and why the mirror can put it back.
Named alongside it
The objects these essays reach for when they reach for this one.
FisheyeAnisotropyAzimuthConformalEquirectangularFocal lengthPicture surfaceBeltramiCanopy photographyCatoptricCatoptric anamorphosisCritical angle