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

Calibration — where it appears

The camera's own numbers: a focal length, a principal point, and whatever else is needed to turn a position in pixels into a direction in space. Everything metric a picture supports is bought with it, and an uncalibrated picture supports projective statements only.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

the eyecorrect from 9 cm, at 160 mm wideoutlines agree to 6e-12 px

One picture of a ball

The outline of a ball in a photograph fixes the cone of rays that graze it, so the cone's axis is the direction of the ball's centre and its half-angle is the arcsine of radius over distance — both exactly, and neither of them separately. A ball a hundred and thirty-seven times larger, a hundred and thirty-seven times further away, draws the identical conic. And the drawn ellipse's own centre is not the image of the ball's.

metrology · Ballrecovery 020406012345the baseline between the two eyes (metres)angle between the cone axes (°)42.6°two cones, one ballcentre to 9e-7°

Two pictures of a ball

Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.

metrology · Ballrecovery -202-400-2000200400along the read stick from its middle (mm)off its own straight line (mm)lean 90°stick 905 mmexact readings, an index of 1.25the bow: 2.216 mm

A straight stick tests the index only across the sight

Told that each stick is straight, three eyes fitting a water surface's height and its index still slide down the valley between them — when the stick leans away from them. A wrong index carries such a stick into another straight stick, off its line by 0.016 mm and 116 mm short. Turned to lean across the line of sight, the same wrong index bows it by 2.2 mm, and the straight-line fit's slide falls from 0.058 to nothing. A tape on the stick, read only to ±50 mm, stops the slide outright.

refraction · Refraction horizoncorrect from 12 cm, at 160 mm wide67° across

The 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.

foundations · Duality

Named alongside it

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

Angular sizeDemonstrationQuadricReconstructionscale ambiguityTangent coneTriangulationAbsolute conicBaselineBiascentre of projectionCollinearity

All concepts