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

Discriminant — where it appears

The quantity B² − 4AC of a conic's coefficients, whose sign names the conic as an ellipse, a parabola or a hyperbola.

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

horizonthe eye stands 8.78 m from the centrecorrect from 26 cm, at 160 mm wideellipse · nearest point +3.96 m

The conic a circle becomes

A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.

foundations · imagedcircle horizon25withheldfive marks fitted, one withheldcorrect from 26 cm, at 160 mm widethe withheld mark is 2e-13 px off the fitted conic

Five marks and the sixth

Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.

foundations · fiveconic

Named alongside it

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

CircleConicpoint at infinityProjective mapcentre of projectiondegrees of freedomdesign matrixDualityEigenvaluesEllipseHomographyHorizon

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