Degenerated Conic Sections

Fig. 10.8 is a diagram of a double cone (two cones meeting at a common vertex V) with a cutting plane passing through that vertex. The plane is tilted so that the angle it makes with the cone’s axis — call this angle $\beta$ — is larger than the cone’s semi-vertical angle $\alpha$, but still less than or equal to $90^\circ$. In other words, $\alpha < \beta \leq 90^\circ$.

The key visual: the plane slices the cone only at the single point V. It does not cut through any other part of the cone’s surface. So the intersection — the “section” — is just that one point. The diagram marks this point in indigo at V, and the two angles $\alpha$ and $\beta$ are labelled to show the relationship.

What this teaches is the idea of a degenerate conic section. A non-degenerate conic (circle, ellipse, parabola, hyperbola) is formed when the plane cuts through the cone away from the vertex. But when the plane passes exactly through the vertex, the intersection shrinks to a simpler shape. Here, because the plane is steeper than the cone’s side ($\beta > \alpha$), it touches only the vertex — giving a single point. This is the degenerate case of an ellipse (or a circle, if $\beta = 90^\circ$).

The textbook uses this figure to ground the concept that conic sections are not always curves; they can collapse into points, lines, or pairs of lines. The formulas that follow in the chapter define each conic by a geometric property (like distance from a focus), but the degenerate cases remind us that those definitions have edge cases.

Fig. 10.9 is a three-dimensional sketch of a double cone (two cones meeting at a common vertex V) with a cutting plane passing through V. The plane is tilted so that it exactly contains one of the straight lines (generators) that lie on the surface of the cone. That particular generator is drawn as a bold indigo line. Two angles are marked at the vertex: $\alpha$, the semi-vertical angle of the cone (the angle between the cone’s axis and any generator), and $\beta$, the angle between the cutting plane and the cone’s axis. In this figure, $\beta = \alpha$.

The physical idea is simple: when a plane cuts through the vertex of a double cone, the intersection is no longer a smooth curve like a parabola, ellipse, or hyperbola — it degenerates into something simpler. If the plane is tilted so that it just grazes the cone along one of its generators (i.e., the plane contains that generator), the intersection is exactly that straight line. The textbook calls this the degenerate case of a parabola because a parabola is the conic you get when $\beta = \alpha$ but the plane does not pass through the vertex. Here, because the plane goes through V, the parabola collapses into a single line.

The key relation is the condition for degeneracy: $\beta = \alpha$. There is no separate formula for the line itself in this figure — the textbook uses the geometry to set up the idea that a conic section can shrink to a point, a line, or a pair of lines when the cutting plane passes through the vertex. The actual equations for the non-degenerate conics come later, but this figure establishes the boundary case: when the plane contains a generator, the section is a straight line.

Fig. 10.10 shows the geometric situation that produces a degenerate hyperbola — a pair of intersecting straight lines. The figure has two panels, (a) and (b), each built around the same double cone.

In both panels, the cone is drawn as two identical nappes meeting at the vertex $V$. A plane passes through $V$ and cuts through both nappes. The key difference between the two panels is the angle the plane makes with the axis of the cone.

Let $\alpha$ be the semi-vertical angle of the cone (the constant angle between the axis and any generator — a straight line on the cone’s surface). Let $\beta$ be the angle between the cutting plane and the axis.

Panel (a) is labelled $0 = \beta < \alpha$. Here the plane is horizontal — it makes an angle $0^\circ$ with the axis, so it is perpendicular to the axis. Because $\beta = 0$ is less than $\alpha$, the plane cuts through the vertex and slices both nappes. The intersection is a pair of straight lines that cross at $V$. These lines are the two generators of the cone that lie in the cutting plane.

Panel (b) is labelled $0 < \beta < \alpha$. The plane is now tilted, but still passes through $V$ and still makes an angle $\beta$ that is smaller than $\alpha$. Again, the plane cuts both nappes, and the intersection is a pair of straight lines through $V$. The lines are different from those in panel (a) — they are the two generators that happen to lie in this particular tilted plane.

In both cases, the intersection is not a smooth curve like a hyperbola; it is two straight lines that cross at the vertex. This is why the textbook calls it a degenerated case of a hyperbola — a hyperbola normally has two separate branches that never meet, but here the two branches have collapsed into straight lines that intersect.

The physical idea the figure teaches is this: a hyperbola is the curve you get when a plane cuts both nappes of a cone but does not pass through the vertex. When you slide that plane down so it passes through the vertex, the two separate branches of the hyperbola shrink into two straight lines that cross at the vertex. The condition $0 \leq \beta < \alpha$ is the mathematical description of when this happens.

The textbook does not derive a formula directly from this figure — the figure is purely geometric. However, the condition $0 \leq \beta < \alpha$ is the foundation for the standard equation of a hyperbola. Later in the chapter, the standard form of a hyperbola is given as: