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

Refraction — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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%

    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.

    part 1 · refraction
  2. water, n = 1.333eyethe point, 1.50 m downsagittal: 0.740 mtangential: 0.320 mno single viewpoint — the rays miss by 0.840 mbetween the pencil's two images

    A point under water has two depths

    The apparent depth of a submerged point is not one number even along one line of sight. The rays it sends to an eye pass through two focal lines, and at 60° from the vertical a point 1.50 m down has an image 0.740 m down and another 0.320 m down. Two eyes side by side read the first, a head moving up and down reads the second, and a pair of eyes tilted between them reads neither — their two rays miss each other by as much as 6.39 mm.

    part 2 · refraction
  3. eyethe sticktwo eyes side by sidea nodding headcorrect from 18 cm, at 160 mm widekink 14.96° · turn 4.29°

    A straight stick in water is a kink and a curve

    The bent stick is described as one kink at the surface. Traced point by point, the image two level eyes see leaves the surface 14.96° off a stick leaning 30° — the same whichever way it leans — and then keeps turning, by 1.11° when it leans away from the eye and 7.62° when it leans toward it. A photograph from the eye shows the kink and almost none of the curve, 1.63 px over 166 px, and when the stick leans straight toward or away from the eye it shows neither: the picture is one straight line.

    part 3 · refraction
  4. eyethe stickcorrect from 18 cm, at 160 mm widerays miss by 3.13 mm

    The stick a stereo pair puts back

    Two eyes side by side reconstruct a submerged stick exactly as the sagittal image — kinked 14.96° — and two eyes one above the other exactly as the tangential one, kinked 9.59°. Roll the baseline between them and the two rays to a point miss each other by up to 3.13 millimetres, past the 2.85 a pixel covers at that range, and the reconstruction is a third stick that is neither — 551 millimetres of a one-metre stick, with its tip at 0.405 metres against a true 0.866.

    part 4 · refraction
  5. the stickreconstructedfitted straightbaseline rolled 45°residual 0.2 mm

    The residual does not warn

    Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.

    part 5 · refraction
  6. 00.50011.502020406080roll of the baseline, degreesmillimetresunweighted residualfloored at a pixelweighted, no floorthe rays' miss, rmsa one-metre stick, the baseline rolledthe weighting runs the wrong way harder

    A fit weighted by the miss trusts only the surface

    Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.

    part 6 · refraction

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