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

The real instrument

A lens is a departure from the pinhole, and the departure is the largest systematic error in every measurement made here. It bends straight lines, destroys the invariant, and can be recovered from nothing but the knowledge that some edges were straight — which is the round trip again, on a harder problem.
principal pointk₁ = -0.320, k₂ = 0.110 — barrel distortioncentre line 0e+0 px of sag, outermost 17.8 px

Straight lines that are not

Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.

 00.50011.502-0.400-0.20000.200k₁departure of the cross-ratio from the world's value (%)pinhole1.29%the pinhole's own error, on the same four points2e-16 — the control

A lens destroys the invariant

The cross-ratio is the one thing a projection preserves, and everything checkable on this site is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.

 fitted k₁ = -0.280000true -0.280000, off by 5e-15

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.

 level — the top is cut off0.00° of spreadtilted 13°4.55° of spreadshifted 95 px0.00° of spreada shift moves every point by exactly the shift95.0 px, and no direction at all

The principal point is not the centre

Every textbook that computes a focal length from two vanishing points supplies the middle of the frame as the principal point. On a shifted or cropped picture that is wrong, and it costs one and a half per cent of the answer at a fifth of a frame's shift.

 the far field — where the stitch was fitted2.2 m — 3.2 px out24 m — 0.3 px outthe sky registers to 1e-13 pxthe foreground does not — up to 3.2 px

The eye is a place, not a point

Rotate a camera about the wrong point and the sky still stitches perfectly while the foreground slides. The misregistration falls as one over the distance, exactly — which is what says the fault is the pivot and not the lens.

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