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

Chasles theorem — where it appears

The result that four points of a conic subtend the same cross-ratio at every fifth point of that same conic. It is what makes a conic a projective object rather than a metric one, and it survives photography exactly, because a projection preserves cross-ratio.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as pascal line — the same set of essays touches all of them, so they are one junction rather than several.

horizon1234the fifth pointcorrect from 23 cm, at 160 mm widecross-ratio 1.627695 at position 1 of 22

Four points on a conic look the same from anywhere on it

Four marks on a photographed circle subtend the same cross-ratio at every point of the curve — 1.627695, unmoved over twenty-two positions of the fifth point, to 1.1e-13 degrees of projective spread. A fifth point 6.1 pixels off the conic reads anything from 1.00 to 2.52, so the invariant belongs to the curve rather than to the four marks.

foundations · Chasles 123456Pascal · family at 0.500correct from 23 cm, at 160 mm widethree meets, collinear to 1.2e-12 px

Pascal's line, and the theorem underneath Pappus

Six points of a conic, three meets of opposite sides, collinear to 1.3e-12 px. Flatten the conic towards a pair of lines and the residual never leaves the axis — 5.7e-14 px at the end, where the theorem has become Pappus's — so Pappus is the degenerate case of Pascal reached continuously rather than a separate theorem that resembles it.

foundations · Fiveconic

Named alongside it

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

ConicPascal lineProjective invariantComplete quadrangleConic fitCross-ratioDegenerate familyDemonstrationDualityImaged circleIncidenceinstrument limit

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