Conic SectionsSections of a Cone

A torch only throws light forwards, so its beam is half a cone. Mathematicians use the whole thing. Start with a fixed vertical line and a second line that crosses it at a fixed point V, making an angle with it.

Now spin round , keeping the angle the same all the way round. The line sweeps out a shaped like two ice-cream cones joined at their tips: a double-napped right circular cone. From now on we just call it a cone. It goes on for ever in both directions.

The point V is the vertex and the line l is the of the cone. Each position of the spinning line m is a generator, a straight line lying right on the surface. The vertex splits the cone into two matching halves, the two nappes — that is what double-napped means.

Fig. 10.1–10.3 — the cone seen from the side, with a cutting plane seen edge-on.

Now take a flat plane and cut the cone with it. The curve where they meet is a conic section. Which curve you get depends on two things: the angle that the plane makes with the axis, and whether or not the plane passes through the vertex.

10.2.1 · Circle, Ellipse, Parabola and Hyperbola

First let the plane miss the vertex. The generators make with the axis.

Tilt the plane and cut all four curves out of the same cone.

${b}°

Here is what you found, for a plane that cuts a nappe anywhere except at the vertex:

(a) When , the section is a (Fig. 10.4).

(b) When , the section is an (Fig. 10.5).

In each of these three cases the plane cuts right across one nappe.

(d) When , the plane is steep enough to cut both nappes, and the section is a (Fig. 10.7).

Note: one family, not four

All four curves come from cone; only the tilt of the knife changes. A level cut gives a circle. Tip the plane a little and the circle stretches into an ellipse. Tip it until it is parallel to a generator and the curve can no longer close: a parabola. Tip it further still and it strikes both nappes, giving the two branches of a hyperbola.

10.2.2 · Degenerated Conic Sections

Now make the plane pass through the vertex V itself. The curves collapse into something much plainer.

Find all three.

${b}°

When the plane cuts the cone at its vertex:

(a) When , the section is a single (Fig. 10.8) — all that is left of a circle or an ellipse.

(b) When , the plane contains a generator and the section is a line (Fig. 10.9). It is the degenerated case of a parabola.

In the rest of the chapter we define each conic by a geometric property instead of by cutting a cone — and that definition hands us its equation.

Speed Round: Name the Cut

You are told the angle of the cone, the angle of the plane, and whether the plane passes through the vertex. Name the section.

Get five right to move on.

α${al}β${be}

${ask}

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Glossary

Conic SectionsSections of a Cone

10.2.1 · Circle, Ellipse, Parabola and Hyperbola

Note: one family, not four

10.2.2 · Degenerated Conic Sections

Speed Round: Name the Cut