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

Principal point — where it appears

7 essays name this object, across 3 fields. What follows is each of them, and the objects they name alongside it.
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.

lens · distortion 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.

lens · distortion recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across

Recovering the camera from the picture it drew

Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.

foundations · recovery 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.

lens · plumbline 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.

lens · shift centric pointdistance point, 12 px off the sheet →430 pxcorrect from 10 cm at 160 mm wide47° across

The distance point is the viewing distance, drawn

There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.

construction · alberti 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.

lens · pupil

Named alongside it

The objects these essays reach for when they reach for this one.

Focal lengthResidualVanishing pointCamera calibrationHomographyHorizonPicture planeRadial distortionsingle-view metrologyBarrel distortionBrown–Conradycentre of projection

All concepts