Plumbline — the series

1. Fitting a lens from straightness alone

   No calibration target, no known scene, no camera. Only the knowledge that some edges in the picture were straight — and the coefficient comes back to fifteen digits. Then it comes back with a companion, and the two are correlated at −0.997.

   part 3 · lens

2. The lines that calibrate a lens

   One straight edge through the centre of a picture says nothing about a lens's distortion, and one 180 px from the centre determines k₁ to 9.0 × 10⁻⁴ — the precision rises in proportion to the offset. But distance from the centre is not enough. Crowd three edges on one side and, the moment the distortion centre is also unknown, the coefficient is ten times worse, because a bend on one side looks like a moved centre; put one edge across the centre and it barely changes.

   part 4 · lens

3. The response is at the ends and the information is not

   A radial map bows a straight edge by an amount that grows as the square of the distance along it, so 93 per cent of an edge's response to the coefficient lies in its outer quarters. Spending the marks there is 16 per cent worse than spreading them evenly, because two clusters say nothing a shifted, tilted line could not say. What identifies the coefficient is a curvature, which needs three places — both ends and the middle, which beats an even spread by 11 per cent.

   part 5 · lens

4. No design separates the two coefficients

   Separating a squared term from a fourth-power one was supposed to need marks at radii far apart, which is a statement about where edges are placed. It is not: one straight edge already runs from 20 px to 326. Spreading ninety-six marks over three edges or sixteen changes the answer by 23 per cent, spreading the offsets makes it 19 per cent worse, and the correlation stays at −0.98 whatever is done. What a plumb-line calibration determines is one number, to 1.52 thousandths, and which number depends on the model.

   part 6 · lens

5. A known target sharpens the fit and does not separate it

   Printed circles of stated size were supposed to break the −0.98 correlation between a lens's two radial coefficients, because a circle puts every mark at one radius and no straight edge can. They do not: every design of circles leaves the pair 0.979 to 0.9997 correlated, and for circles of known size the figure is exactly the cosine between r³ and r⁵. What a known target buys is precision — 3.8 times the straight edges' at the same budget — and only if its size in the picture is known to about a thousandth.

   part 7 · lens

All series