Series

Planar — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. left pictureright pictureevery one of them fits every mark1.6e-6 px

    A flat scene fixes no second eye

    Eight marks on one plane leave the eight-point design matrix two ranks short, so a two-parameter family of fundamental matrices satisfies every mark exactly — three of its members are 1.41 apart after normalisation and all three fit to 1.6 × 10⁻⁶ pixels. The same photographs determine the plane's own map from four marks, to 5.9 × 10⁻¹³ pixels.

    part 1 · twoviews
  2. 0306090-3-2-1out-of-plane spread, as a fraction of the scene's own extent (powers of ten)error in the recovered translation direction, in degreeswith 0.3 px of reading errorexact marksthe geometry is a step and the measurement is a slope9 reliefs

    How flat is flat enough

    With exact marks the transition has no width at all — 84° of pose error at exactly coplanar and 0.000° at eight parts in ten thousand of relief. Put three tenths of a pixel of reading error in and the same sweep becomes a slope three decades wide, crossing into usefulness when the out-of-plane parallax reaches about ten times the marking error.

    part 2 · twoviews
  3. the second pictureepipole22 parallax lines miss the epipole by at most 1.7e-10 pxsecond camera stepped forward

    Two marks off a known plane find the other eye

    Map a courtyard's ground from one picture into the other, and every raised mark lands somewhere the map did not send it — displaced along a line through the image of the other camera, to a fifth of a billionth of a pixel. Two such marks put that image where it is, and with it the whole epipolar geometry.

    part 3 · twoviews

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