Double Cone — Definition, Formula & Examples
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.
Double Cone
Key Formula
z2=x2+y2
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.
z2=x2+y2
Step 2: Substitute the coordinates x = 3, y = 4, z = 5 into the left side of the equation.
z2=52=25
Step 3: Substitute into the right side of the equation.
x2+y2=32+42=9+16=25
Step 4: Compare both sides. Since 25 = 25, the equation is satisfied.
25=25✓
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.
62=x2+y2
Step 2: Simplify the left side.
36=x2+y2
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=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 z2=x2+y2 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