The thread: Fitted, not assumed — page 5
Which rule a fisheye obeys, from straightness alone
Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.
What survivesA floor with a referent
Recover a focal length from two vanishing points and measure more points along each line. Through a pinhole the error falls from 0.34 per cent to 0.05 and the instrument finds no floor at all. Through a lens of k₁ = −0.05 it falls, turns, and rises to 0.70 per cent — because the noise the extra points removed had been partly masking the lens's bend. The floor is 0.72 per cent of the focal length, and doubling the distortion coefficient doubles it to 1.44. It is not noise and not conditioning; it is the model, priced.
Light and mirrorsThe arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
What survivesThe horizon has a pole
Every other duality in this collection is a matrix somebody picked. A calibrated camera fixes one nobody picked, and under it the horizon of a plane and the vanishing point of that plane's normal are pole and polar — to 2.7 × 10⁻¹² pixels. Run backwards, those two marks give the focal length with no known length, no right angle and no square anywhere in the scene.
What survivesFive tangents name the same conic
Fitting a conic to five lines is the same six-coefficient nullspace problem as fitting one to five points, with the roles exchanged, and it lands on the same curve to 2.6 × 10⁻¹¹ pixels. What duality does not promise is that the two are equally good evidence — and measured at equal marking precision they are, to within seventeen per cent, because a drawn tangent is made of points.
Light and mirrorsThe lamp's size over its distance, and nothing else
One straight-edge shadow gives a lamp's angular size and nothing about its actual size or distance — scale a 36 cm lamp and its 3.0 m distance together and the band it casts differs by 0.0e+0. A second card at a different height breaks the tie, recovering 36.00 cm at 3.000 m from bands of 18.0 cm and 144.0 cm alone, at a condition number of 37.1.
Light and mirrorsThe lamp a low shadow cannot locate
Five posts, their shadows, and the line from each foot through its own shadow's tip meet at the lamp standing over them — at 52 degrees up that meet moves 0.11 m for half a pixel of marking error; at 11.5 degrees, 0.64 m. The two eigenvalues of the same pencil of lines part by a factor of 8849084 across the sweep, and half a pixel becomes more than a metre of lamp below 14.4 degrees.
Light and mirrorsA shadow edge read as a profile
A lamp, a stick and a camera recover a stepped object's profile to 4.9e-15 m rms when the marks are exact, and to 10.4 mm once they are read to two tenths of a pixel — the same linear law a fitted exponent of 1.001 confirms. What actually sets that number is the angle between the sweeping light plane and the camera's own ray — the amplification is least, 17.0 times a pixel, broadside at 6°, and grows without bound toward -36.1°, where the plane contains the camera's own eye and the recovery keeps none of its marks at all.