Ellipse — 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 hyperbola — the same set of essays touches all of them, so they are one junction rather than several.
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.
Two lines at infinity
A picture of a plane has two of them and they are not the same line. One is the horizon, where the plane's own infinity went; the other is where the picture's coordinates run out. The words ellipse and hyperbola are about the second, and every scrap of metric information is on the first — so a circle whose photograph is a hyperbola calibrates exactly as well as one whose photograph is an oval, to 7e-14 of a degree.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicHorizonHyperbolapoint at infinityProjective mapVanishing lineAbsolute coniccentre of projectionCircleCircular pointsDiscriminantIncidence