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

Fiveconic — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. horizon25withheldfive marks fitted, one withheldcorrect from 26 cm, at 160 mm widethe withheld mark is 2e-13 px off the fitted conic

    Five marks and the sixth

    Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.

    rung 3 · foundations
  2. horizonflat on the groundleaning 23.6°two circles, 6.4 m across, in planes 23.6° apartcorrect from 26 cm, at 160 mm widetwo poses, 23.6° apart, one picture

    Two circles, one picture

    A photographed circle leaves its own pose ambiguous, and not a little: two congruent circles in planes 23.6° apart draw the same conic to 1.1e-16 on normalised coefficients, both of them in front of the camera. On top of that the distance is free, so the family is two discrete poses each with one continuous parameter — and a plane one degree from either draws a conic 2.5e-4 away, which is what makes the agreement a measurement.

    rung 4 · foundations

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