Concept
Dandelin spheres
Two spheres inscribed in a cone on either side of a cutting plane, touching that plane at the section's two foci. They turn the definition by a cut through a cone into the definition by two foci, with no algebra anywhere in the argument.
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
One cone, four curves
The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.
Every ray comes back to the other focus
An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicEllipseFocusParabolaDegenerate conicDirectrixEccentricityHyperbolaOptimalityProjectionReflectionTangency