a conic as a plane section of a cone

Why are the parabola, ellipse, and hyperbola lumped together under one name, 'conic sections'? Because the ancient Greeks discovered them all by slicing a single object — a cone — with a flat plane and looking at the curve where the cut meets the surface. Tilt the slicing plane at different angles and the same cone yields all the curves in turn. This is the original, geometric reason the three are siblings.

Take a double cone: two cones joined tip to tip at a common vertex, extending infinitely both ways, like an hourglass with no waist. Now cut it with a plane and watch the edge of the cut. Slice straight across, perpendicular to the axis, and you get a circle. Tilt the plane a little, less steeply than the side of the cone, and the circle stretches into a closed ellipse. Tilt it exactly parallel to one side of the cone, and the curve can no longer close — it becomes a single open parabola. Tilt it steeper still, so the plane meets both halves of the double cone, and you cut two separate open curves — the two branches of a hyperbola. If the plane happens to pass through the vertex itself, you get the degenerate cases: a single point, one line, or a crossed pair of lines.

This picture explains the deep unity: one cone, one moving plane, four kinds of curve sorted purely by the angle of the cut. The classification by eccentricity (e = 0, 0 < e < 1, e = 1, e > 1) is the algebraic echo of this geometric tilting. The marvelous bridge connecting this 'cut the cone' definition to the focus-directrix and focal-distance definitions is supplied by the Dandelin spheres.