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 3 essays across 2 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
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 coneAbsolute conicBaselinecentre of projectionConditioningConicDepth from disparity

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