Conic

circle, ellipse, parabola, hyperbola — one cone, sliced at four angles.

pull the knife through a double cone — the curve is whatever angle you cut

A cone sliced level gives a circle. Tilt the knife and the circle stretches into an ellipse. Keep tilting, and there is exactly one angle — the angle parallel to the cone’s own slanted side — where the far edge of the cut never closes: that is a parabola, the knife-edge case between the other two. Tilt past it and the plane cuts through both halves of the double cone at once, and the curve splits into the two separate arms of a hyperbola. Four names for four stops along one continuous turn of the same knife.

Concepts: geometry, topology.

An interactive you play with one thumb, on Wundr.