Circle — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
Two circles, one picture
A photographed circle leaves its own pose ambiguous, and not a little: two congruent circles in planes 23.6° apart draw the same conic to 1.1e-16 on normalised coefficients, both of them in front of the camera. On top of that the distance is free, so the family is two discrete poses each with one continuous parameter — and a plane one degree from either draws a conic 2.5e-4 away, which is what makes the agreement a measurement.
Named alongside it
The objects these essays reach for when they reach for this one.
Conicpoint at infinityProjective mapcentre of projectiondegrees of freedomDiscriminantEigenvaluesPicture planesingular valuesdesign matrixDualityEllipse