Series

Slab — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. no single viewpoint — the rays miss by 4.5 px, depth-dependentf recovered from it: 396.88 px

    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.

    part 2 · refraction
  2. 00.2000.4002468how far off the perpendicular the sightline is (°)how far the pane moves the point (mm)13.2 mm, n = 1.3510.0 mm, n = 1.528.0 mm, n = 1.756.5 mm, n = 2.1t(1 − 1/n) = 3.421 mm for all four1.8 µm apart over 8°

    A pane gives a product before it gives two numbers

    A flat pane of glass displaces every point it is seen through, and the displacement at small angles is the thickness times one minus the reciprocal of the index. So the two numbers arrive multiplied together. Four panes from 6.5 to 13.2 millimetres thick, with indices from 1.35 to 2.1, agree to under two microns over an eight-degree fan and separate by more than a millimetre over sixty — and a fit over the narrow fan returns whichever pair it started near.

    part 3 · refraction
  3. x axis, 54.7° off1.58°y axis, 63.5° off3.39°z axis, 46.9° off1.45°the frame's best rotation1.14°the recovered camera's turn3.37°wedge 2° · focal 1.33% short · principal point 16.6 pxframe mispredicted by 14.7 px

    A wedge of glass turns the camera behind it

    A pane with parallel faces moves every point and no direction, so the camera recovered through a window is the camera that took the picture. Tilt one face 2° and every direction turns, by 1.04° on the axis and 1.67° forty degrees off it. The best rotation of the frame, 1.14°, still leaves 0.94 px, and no homography does much better, so the picture is no longer a projection from the camera's centre. The camera recovered from three vanishing points through the same glass is turned 3.37° — three times as far — because vanishing points lie where the glass bends most.

    part 4 · refraction
  4. 012204060how far off the axis the read directions reach, in degreeshow far the recovered wedge angle is out, in degreesthe wedge's own angle — nothing recovered60 trials at each spread, half a pixel of reading error0.068° at 55°

    The wedge recovered with the camera

    Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.

    part 5 · refraction

All series