Duality — where it appears
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as tangent — the same set of essays touches all of them, so they are one junction rather than several.
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.
The polar with a straightedge
Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicProjective mapTangentcentre of projectionCircleComplete quadrangleCross ratiodegrees of freedomdesign matrixDiscriminantEigenvaluesHarmonic conjugate