Geometric Description of a Conic Section

Let the double-napped cone have its axis vertical, and consider a cutting plane that does not pass through the vertex, intersecting only one nappe:

• A plane perpendicular to the axis (plane $C$) yields a circle.
• A plane tilted, so it is no longer perpendicular to the axis but still crosses only one nappe, yields an ellipse (plane $E$).
• A plane parallel to a side (generator) of the cone yields a parabola.
• A plane parallel to the axis itself (equivalently, parallel to the plane containing the axis), crossing both nappes, yields a hyperbola — which is why the hyperbola alone has two separate branches: one from each nappe. …