Sections of a Cone

Fig. 10.1 is a schematic that builds the definition of a right circular cone from two lines. The fixed vertical line $l$ (drawn with a double arrowhead to show it extends infinitely in both directions) is the axis of the cone. The slanted line $m$ (often shown in a different colour, like blue) is the generator — it will sweep out the cone's surface. The two lines intersect at point $V$, the vertex. The acute angle between $l$ and $m$ at $V$ is labelled $\alpha$; this is the semi-vertical angle of the cone.

The physical idea is simple: imagine holding a fixed vertical rod (line $l$) and a second rod (line $m$) hinged at $V$ so that the angle between them stays exactly $\alpha$. If you rotate $m$ around $l$ like a spinning skipping rope, the path traced by $m$ in space is the surface of a double-napped right circular cone. The word "double-napped" means the cone extends both above and below $V$ — two identical cones meeting at the vertex, like an hourglass shape. Every position of $m$ during this rotation is called a generator of the cone.

This figure is the foundation for classifying conic sections. Once the cone is defined, the textbook introduces a second angle $\beta$: the angle that a cutting plane makes with the vertical axis $l$. The type of curve you get — circle, ellipse, parabola, or hyperbola — depends entirely on how $\beta$ compares to $\alpha$.

Every symbol in that rule comes directly from Fig. 10.1 and its companion Fig. 10.3: $\alpha$ is the fixed angle between $l$ and $m$ at $V$, and $\beta$ is the angle between the cutting plane and the axis $l$. When the plane is horizontal ($\beta = 90^\circ$), you get a circle. As the plane tilts, the curve stretches into an ellipse, then a parabola when the plane is exactly parallel to a generator ($\beta = \alpha$), and finally a hyperbola when the plane cuts both nappes ($\beta < \alpha$). …

The figure shows a double-napped right circular cone — the central object from which all conic sections are born. At the heart of the diagram is the point V, the vertex, where the two halves of the cone meet. A straight vertical line labelled l passes through V and runs the full height of the figure; this is the axis of the cone. Slanting away from the axis is a second straight line labelled m, the generator. The generator makes a fixed angle $\alpha$ with the axis at V.

The cone itself is drawn as two elliptical rims — one at the top, one at the bottom — connected by two straight generator lines that cross at V. The upper half of the cone, above V, is the upper nappe; the lower half, below V, is the lower nappe. The generator m is shown in one particular position, but the idea is that m sweeps around the axis l, always keeping the same angle $\alpha$, to trace out the entire conical surface.

The physical idea the figure teaches is simple: a right circular cone is formed by rotating a line (the generator) around another line (the axis) at a fixed angle. The vertex is the only point where the two nappes meet. Every point on the surface of the cone lies on some generator, and every generator makes the same angle $\alpha$ with the axis.

From this figure, the textbook develops the key relationship that governs all conic sections. When a plane cuts the cone, the angle $\beta$ that the plane makes with the vertical axis determines which curve you get. The central formula is the condition that relates the cutting angle $\beta$ to the cone's fixed angle $\alpha$:

Fig. 10.3 is the key diagram that turns the abstract definition of a cone into the concrete geometry that produces each conic section. It shows a double-napped cone — the full, two-sided surface generated by rotating a line $m$ around a vertical axis $l$ — and a single, tilted plane cutting through it.

The plane is drawn as a blue parallelogram, a visual convention that makes its orientation clear without cluttering the diagram. It slices through only the upper nappe of the cone, not the lower one. At the vertex $V$, two angles are marked: $\alpha$, the constant angle between the generator $m$ and the axis $l$, and $\beta$, the angle between the cutting plane and the axis $l$. The labels "Plane" and "Cone" are placed directly on the respective surfaces.

The physical idea is simple but powerful: the shape of the curve where the plane meets the cone depends entirely on how $\beta$ compares to $\alpha$. If the plane is nearly horizontal ($\beta$ close to $90^\circ$), the intersection is a circle. Tilt it a little ($\beta > \alpha$ but not $90^\circ$), and you get an ellipse. Tilt it exactly so the plane is parallel to a generator ($\beta = \alpha$), and the curve becomes a parabola. Tilt it even steeper ($\beta < \alpha$), and the plane cuts both nappes, producing a hyperbola.

The textbook develops no single formula directly from this figure, but it establishes the geometric foundation for the standard equations that follow. The relationship between $\alpha$ and $\beta$ determines the eccentricity $e$ of the conic, which later appears as the ratio of distances from a point on the curve to a fixed point (focus) and a fixed line (directrix). For a right circular cone, the eccentricity is given by:

where $\alpha$ is the semi-vertical angle of the cone (the angle between the axis and a generator), and $\beta$ is the angle between the cutting plane and the axis. When $\beta = \alpha$, $e = 1$ (parabola); when $\beta > \alpha$, $\cos \beta < \cos \alpha$, so $e < 1$ (ellipse); when $\beta < \alpha$, $\cos \beta > \cos \alpha$, so $e > 1$ (hyperbola). The circle corresponds to $\beta = 90^\circ$, giving $e = 0$. …