Sections of a Cone

A conic section (or simply a conic) is any curve obtained by intersecting a plane with a double right circular cone -- two identical cones placed apex to apex (called nappes), extending indefinitely in both directions along a common axis. Every cone of this kind has a fixed semi-vertical angle $\alpha$, the angle between the axis and any generator (a straight line lying on the surface of the cone and passing through the vertex $V$). The shape of the curve cut out depends entirely on the angle $\beta$ that the intersecting plane makes with the axis of the cone, compared with $\alpha$.

Circle. When the cutting plane is perpendicular to the axis (so $\beta=90^\circ$) and does not pass through the vertex, the section is a circle.

Ellipse. When the plane is tilted, cutting only one nappe completely, so that $\alpha<\beta<90^\circ$, the section is an ellipse -- a circle is, in fact, the special case $\beta=90^\circ$ of this family.

Parabola. When the cutting plane is tilted until it becomes parallel to exactly one generator of the cone (so $\beta=\alpha$), and still meets only one nappe, the section is a parabola -- an unbounded, open curve.

Hyperbola. When the plane is tilted further still, so that $0\le\beta<\alpha$, it cuts both nappes of the double cone, producing a curve with two separate branches, called a hyperbola.

Degenerate conic sections. All four curves above arise from a plane that misses the vertex $V$. If the cutting plane is instead made to pass through the vertex itself, three degenerate (limiting) cases replace the ordinary conics:

• If $\beta=90^\circ$ (plane perpendicular to the axis, through $V$), the plane touches the cone only at $V$ itself: the section degenerates to a single point (the vertex) -- the limiting case of an ellipse/circle shrinking to zero size.
• If $\beta=\alpha$ (plane parallel to a generator, through $V$), the plane contains exactly one generator: the section degenerates to a single straight line -- the limiting case of a parabola.
• If $0\le\beta<\alpha$ (plane through $V$, steep enough to meet both nappes), the plane contains two distinct generators, one from each nappe: the section degenerates to a pair of intersecting straight lines through $V$ -- the limiting case of a hyperbola.

These degenerate cases confirm that the circle, ellipse, parabola and hyperbola are not four unrelated curves but four members of one family, all generated the same way, differing only in how steeply the cutting plane is tilted relative to the cone.