Q:

# What is a good way to go about classifying conic sections?

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The intersection of a cone and a plane results in a circle, an ellipse, a parabola or a hyperbola. In the special case where the plane intersects the vertex of the cone, the resulting section is a point, a line or two intersecting lines.

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When defining conic sections, mathematicians assume that the cone is a doubly infinite cone, meaning two cones that meet at a common apex and extend symmetrically on both sides of the apex infinitely. An ellipse is formed when the intersection of cone and plane results in a closed curve. A circle is a special kind of ellipse formed when the plane intersects the cone perpendicular to the symmetrical axis of the cone. If the plane intersects the cone parallel to one generating line of the cone, the resulting section is a parabola. If the plane cuts through the cone parallel to the symmetrical axis of the cone, the resulting section is a hyperbola, two unbounded curves intersecting both halves of the doubly infinite cone. In the special case where the plane intersects the cone at the apex but not parallel to either generating line of the cone, the result is either a single point or two intersecting lines. If the plane is parallel to either generating line and passes through the apex, the result is one straight line.

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