Visual Plane
every picture states the point it is correct from
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The thread: Not a projection at all

Every theorem quoted on this site is a theorem about a map through a centre. Water is not one; a real lens is not one. Rather than mention that and move on, these essays measure what survives and what does not — a cross-ratio 1.2% out where the pinhole is exact to the last bit, a bundle of rays that misses its own centre by ten millimetres, and the two things that survive anyway.
principal pointk₁ = -0.320, k₂ = 0.110 — barrel distortioncentre line 0e+0 px of sag, outermost 17.8 px The real instrument

Straight lines that are not

Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.

 water, n = 1.333eyetruly 1.50 m downappears 0.818 m downh/n would be 1.125 mno single viewpoint — the rays miss by 17.1° of bend at the surfaceapparent depth 54.5% of the true one, not 75.0% Through water and glass

What a ray does at a surface

A pool looks three-quarters as deep as it is — but only if you look straight down. At sixty degrees the same bottom appears at half its depth, and at eighty at a fifth, which is why the far end of a pool looks shallow enough to walk in.

 00.50011.502-0.400-0.20000.200k₁departure of the cross-ratio from the world's value (%)pinhole1.29%the pinhole's own error, on the same four points2e-16 — the control The real instrument

A lens destroys the invariant

The cross-ratio is the one thing a projection preserves, and everything checkable on this site is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.

 the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m Through water and glass

A picture through water has no viewpoint

Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.

 no single viewpoint — the rays miss by 4.5 px, depth-dependentf recovered from it: 396.88 px Through water and glass

What survives a pane of glass

A slab of glass moves every point of a picture and moves no direction at all. So the camera recovered from a photograph taken through a display case is exactly the camera that took it — out of a picture in which nothing is where it was.

 15°30°45°60°85°the rim: 48.61° from straight upbeyond it: the bottom, reflected45° of sky0.469460° of sky0.370275° of sky0.211485° of sky0.0738area scalen = 1.333, so the rim is at asin(1/n) = 48.61°area scale 0.563 at the centre, 0.0738 at 85° Through water and glass

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.

 00.50011.50205101520the dome's centre, off the entrance pupil (mm)worst departure from the pinhole it would be in air (degrees)centred: exactly zero6 mm → 0.635°a 100 mm dome in acrylic, n = 1.4910.106° per mm of centring error Through water and glass

The port that is not there

A flat window into water costs a lens a third of its field. A sphere centred on the entrance pupil costs nothing at all — not nearly nothing, exactly nothing — and six millimetres off centre costs 0.635°.

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