Sections of a cone

A right circular cone has a vertex $V$, a straight axis through $V$, and a slanted surface swept out by a line (the generator) that touches the vertex and makes a fixed angle $\alpha$ with the axis. The "double-napped" cone is two such cones joined tip to tip, extending without limit in both directions. Cut this solid with a plane that makes angle $\beta$ with the axis, and keep the plane away from the vertex — the shape of the cut depends only on comparing $\beta$ with $\alpha$.

• Circle: the plane is perpendicular to the axis, $\beta = 90^\circ$. This is the special, most symmetric case of an ellipse.
• Ellipse: the plane cuts only one nappe, and $\beta > \alpha$ (but is not perpendicular to the axis). The section is a single closed, bounded curve.
• Parabola: the plane is parallel to exactly one generator of the cone, $\beta = \alpha$. The curve is open — it never closes up.
• Hyperbola: the plane cuts both nappes of the double cone, $\beta < \alpha$. The two nappes each contribute one branch, so the curve has two separate, mirror-image pieces.

These four are the non-degenerate conics. If the cutting plane is allowed to pass through the vertex itself, the "sections" collapse to a single point, a single line, or a pair of intersecting lines — the degenerate conics — which are a separate, much smaller topic and are not covered here. Every non-degenerate conic can also be described without any reference to a cone at all: it is the locus of a point $P$ whose distance from a fixed point (the focus $S$) stays in a constant ratio, the eccentricity $e$, to its distance from a fixed line (the directrix):

where: $S$ is the focus, $P$ is any point on the curve, and $e$ is a fixed positive number — $e = 1$ gives a parabola, $0 < e < 1$ gives an ellipse (with $e = 0$ the limiting case of a circle), and $e > 1$ gives a hyperbola. This focus–directrix definition is the bridge to the coordinate equations of the individual conics studied later in the chapter; at this stage, the cone picture above remains the primary way to recognise which conic a given cut produces.

Picture standing at the vertex $V$ and looking along a single generator line, tilted at angle $\alpha$ to the axis; the whole cone is just this line spun once around the axis. A cutting plane meets the cone in a closed or open curve depending on whether it eventually runs parallel to that spinning generator or not, and this is why the comparison between $\beta$ and $\alpha$ is the whole story. If the plane is steeper than every generator relative to the axis ($\beta > \alpha$), it can never become parallel to any generator as it sweeps around the cone, so it re-enters and closes up on itself after slicing through the one nappe it started in — a bounded, closed curve, the ellipse (or, at $\beta = 90^\circ$, its most symmetric case, the circle). If the plane tilts down until it is exactly parallel to one generator ($\beta = \alpha$), the cutting plane and that one generator never meet again, however far the cone is extended — the section can never close up, so it trails off to infinity along the one direction the plane matches: the parabola. Tilt the plane past that point ($\beta < \alpha$) and it is now shallower than the generator's own slope; it must eventually cross into the other nappe of the double cone as well, because a plane that steep relative to the axis, extended far enough, cannot stay confined to a single nappe. Each nappe then contributes one open, unbounded branch, and the two branches are mirror images of each other about the axis — the hyperbola's two disjoint pieces.

Problem. A double-napped right circular cone has generators making an angle of $50^\circ$ with the axis. A plane, not passing through the vertex, cuts the cone at $50^\circ$ to the axis. Identify the conic, then a second plane at $25^\circ$ to the axis (same cone) is considered — identify that conic too, and explain why it has two branches.

Step 1. For the first plane, the cone's generator angle is $\alpha = 50^\circ$ and the cutting angle is $\beta = 50^\circ$, so $\beta = \alpha$.

Step 2. By the classification, $\beta = \alpha$ with the plane clear of the vertex means the plane is parallel to exactly one generator — the section is a parabola.

Step 3. For the second plane, $\beta = 25^\circ$ against the same $\alpha = 50^\circ$, so $\beta < \alpha$: the plane is shallower than the generator, tilted enough to pass into both nappes of the double cone.

Step 4. By the classification, $\beta < \alpha$ gives a hyperbola. It has two branches because the single plane meets the upper nappe in one curve and the lower nappe in its mirror image; together these two disjoint pieces form the complete hyperbola.

Answer: the $50^\circ$ plane gives a parabola; the $25^\circ$ plane gives a hyperbola with two branches, one from each nappe.

Competitive stretch

A third plane on the same cone ($\alpha = 50^\circ$) is now cut at $\beta = 70^\circ$ to the axis. Since $70^\circ > 50^\circ$, this is the $\beta > \alpha$ case, so it produces an ellipse — a single closed curve confined to one nappe, distinct from both the parabola and hyperbola obtained above from the same cone at different tilts.

• Assuming any tilted plane gives an ellipse — tilt far enough to match the generator and you get a parabola instead, and tilt past that to get a hyperbola, not "a stretched-out ellipse."
• Forgetting "double-napped" means two cones joined at a common vertex, extending infinitely both ways — a single nappe alone can never produce a hyperbola.
• Mixing up the angle $\beta$ with the axis and the angle the plane makes with the base (which is $90^\circ - \beta$); the classification above is stated strictly in terms of the axis angle.
• Forgetting the plane must miss the vertex — sections through the vertex give the separate degenerate cases (a point, a line, or a pair of intersecting lines), not a circle, ellipse, parabola, or hyperbola.

Ask the tutor: Want to see how tilting a single cutting plane, step by step, sweeps through circle, ellipse, parabola, and hyperbola on the same cone?