Double Cone: Two Cones Joined Apex-to-Apex (Conic Sections)

Double Cone

A geometric figure made up of two right circular cones placed apex to apex as shown below. Typically a double cone is considered to extend infinitely far in both directions, especially when working with conic sections and degenerate conic sections.

Note: The graph of the equation z² = x² + y² is a standard way to represent a double cone. That is the equation for the image below.

Two cones joined apex-to-apex along a shared vertical axis, forming an hourglass shape, with dashed ellipses showing the circular bases at top and bottom.

Double Cone

Key Formula

z^2 = x^2 + y^2

Where:

$x$ = Horizontal coordinate in the first horizontal direction
$y$ = Horizontal coordinate in the second horizontal direction
$z$ = Vertical coordinate along the axis of symmetry of the double cone

Worked Example

Problem: Verify that the point (3, 4, 5) lies on the standard double cone given by z² = x² + y².

Step 1: Write down the equation of the double cone.

z^2 = x^2 + y^2

Step 2: Substitute the coordinates x = 3, y = 4, z = 5 into the left side of the equation.

z^2 = 5^2 = 25

Step 3: Substitute into the right side of the equation.

x^2 + y^2 = 3^2 + 4^2 = 9 + 16 = 25

Step 4: Compare both sides. Since 25 = 25, the equation is satisfied.

25 = 25 \quad \checkmark

Answer: Yes, the point (3, 4, 5) lies on the double cone z² = x² + y².

Another Example

This example explores the geometry of the double cone by slicing it at a fixed height, showing that horizontal cross-sections are circles whose radii grow linearly with |z|.

Problem: A double cone has the equation z² = x² + y². Find the radius of the circular cross-section at height z = 6.

Step 1: Set z = 6 in the double cone equation to find the cross-section at that height.

6^2 = x^2 + y^2

Step 2: Simplify the left side.

36 = x^2 + y^2

Step 3: Recognize that x² + y² = 36 is the equation of a circle centered at the origin in the xy-plane with radius r.

r = \sqrt{36} = 6

Answer: The cross-section at z = 6 is a circle of radius 6. Notice that for the standard double cone, the radius of any horizontal cross-section equals |z|.

Frequently Asked Questions

What is the difference between a cone and a double cone?

A single cone has one apex and extends in one direction from that apex, forming a single sheet. A double cone has two identical cones sharing the same apex, extending in opposite directions along the same axis. When mathematicians discuss conic sections, they almost always mean slicing a double cone, not a single one.

Why is a double cone important for conic sections?

Every conic section—circle, ellipse, parabola, and hyperbola—is obtained by slicing a double cone with a flat plane at different angles. A hyperbola, in particular, requires both halves (called nappes) of the double cone, which is why a single cone is not sufficient to generate all conics.

What are the nappes of a double cone?

Each of the two individual cone-shaped halves of a double cone is called a nappe. The upper nappe extends upward from the apex, and the lower nappe extends downward. A plane that cuts through both nappes produces a hyperbola.

Double Cone vs. Single Cone (Right Circular Cone)

	Double Cone	Single Cone (Right Circular Cone)
Structure	Two cones joined apex to apex, extending infinitely in both directions	One cone with an apex and a single base or extending infinitely in one direction
Number of nappes	Two	One
Standard equation (3D)	z² = x² + y²	z = √(x² + y²) (upper nappe only)
Conic sections produced	All conics: circle, ellipse, parabola, hyperbola, and degenerate cases	Circle, ellipse, parabola only (cannot produce a hyperbola)
Common use	Defining and deriving conic sections	Volume and surface area problems in geometry

Why It Matters

You encounter the double cone most directly when studying conic sections in algebra and precalculus—the circle, ellipse, parabola, and hyperbola are all defined as cross-sections of a double cone. Understanding this surface also prepares you for multivariable calculus, where the equation

z^2 = x^2 + y^2

appears as a standard quadric surface. Recognizing how the slicing angle determines which conic you get is a foundational idea in analytic geometry.

Common Mistakes

Mistake: Using z = x² + y² instead of z² = x² + y² for the double cone equation.

Correction: The equation z = x² + y² describes a paraboloid, not a cone. A double cone requires z² on the left side: z² = x² + y². Squaring z is what allows both positive and negative z-values, giving you both nappes.

Mistake: Thinking a single cone is enough to generate all conic sections.

Correction: A hyperbola consists of two separate branches, each lying on a different nappe. You need both nappes—a double cone—for the slicing plane to produce a hyperbola.

Related Terms

Cone — A single nappe of the double cone
Right Circular Cone — The specific cone type forming each nappe
Conic Sections — Curves produced by slicing a double cone
Degenerate Conic Sections — Special cases when the plane passes through the apex
Apex — The shared tip where both cones meet
Geometric Figure — General category that includes the double cone
Graph of an Equation or Inequality — The double cone is the graph of z² = x² + y²
Equation — z² = x² + y² defines the surface algebraically