Hyperbola — 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.
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.
Three conics are one conic and a choice of horizon
Ellipse, parabola and hyperbola are one curve and three answers to which line is infinitely far away. Pitching one camera over one 6 metre circle walks through all three, and the crossing sits at 30.465545° by two instruments with different units — but the line that decides is not the horizon, and the popular name for the choice names the wrong one of the two lines a picture of a plane has.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicEllipseHorizonpoint at infinityProjective mapVanishing lineDiscriminantline at infinityParabolaAbsolute coniccentre of projectionCircle