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

Curvedreceiver — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface

    The floor that is not a plane

    A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest.

    part 4 · light
  2. correct from 20 cm, at 160 mm wideresidual 2e-15 m · 1/sin up to 1.22

    A shadow across a second object

    A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.

    part 5 · light
  3. -200200204060position around the occluder's outlinehow far the four-point map mispredicts, signed (mm)a dished floor (k = 0.289)a ridged floor (k = 0.183)a floor with a step (k = 0.040)matched at 25.00 mm worstdish 0.1% · ridge 2.8% · step 19.4%

    The residual has a shape

    A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.

    part 6 · light
  4. two lamps: what one centre leaves, floor by floora flat floor138.33 pxa dished floor138.33 pxa dished floor, k = 0.02138.33 pxa dished floor, k = 0.06138.33 pxa dished floor, k = 0.12138.33 pxa ridged floor138.33 pxa ridged floor, k = 0.02138.33 pxa ridged floor, k = 0.06138.33 pxa ridged floor, k = 0.12138.33 pxa floor with a step138.33 pxa floor with a step, k = 0.02138.33 pxa floor with a step, k = 0.06138.33 pxa floor with a step, k = 0.12138.33 pxtwo lamps, four floors, four curvaturesspread over all thirteen: 2.8e-14 px

    A floor cannot fake a second lamp

    Cast the same two lamps onto four floors at four curvatures and ask how well one centre explains the drawing. Every one of the thirteen answers is 138.3277 pixels — the same to fifteen digits, because a floor decides where along a ray the shadow's tip landed, and a line through a point and another point that has slid along it is the same line.

    part 7 · light

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