Degenerate conic sections

If the cutting plane passes through the vertex V itself, the curve collapses into something simpler, called a degenerate conic.

α < β ≤ 90°: the plane meets the cone only at V, so the section is a single point.

β = α: the plane touches the cone along one generator, so the section is a straight line. This is the degenerate parabola.

0 ≤ β < α: the plane slices through V and both nappes, and the section is a pair of straight lines crossing at V. This is the degenerate hyperbola.

The same comparison of β with α that sorts the ordinary conics also sorts the degenerate ones: β above α gives the ellipse family, β equal to α the parabola family, β below α the hyperbola family.