Elliptic Cone — Definition, Formula & Examples

An elliptic cone is a three-dimensional surface formed by straight lines that pass through a fixed point (the apex) and trace an ellipse rather than a circle. It looks like a standard cone that has been stretched wider in one direction than the other.

An elliptic cone is a quadric surface generated by a family of lines through a common vertex, whose cross sections perpendicular to the axis are ellipses. In standard position with vertex at the origin, its equation is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \dfrac{z^2}{c^2}$ , where $a \neq b$ .

Key Formula

\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}

Where:

$a$ = Semi-axis length of the elliptical cross section in the x-direction at unit height
$b$ = Semi-axis length of the elliptical cross section in the y-direction at unit height
$c$ = Scaling factor along the vertical (z) axis

How It Works

Imagine taking a circular cone and squeezing it so its round cross section becomes an oval. Every horizontal slice through an elliptic cone produces an ellipse. If you slice at height

z = h

, the semi-axes of that ellipse are

\dfrac{ah}{c}

and

\dfrac{bh}{c}

. When

a = b

, the cross sections become circles and the shape reduces to an ordinary circular cone.

Worked Example

Problem: An elliptic cone has the equation x²/9 + y²/4 = z²/25. Find the semi-axes of the elliptical cross section at height z = 5.

Identify parameters: From the equation, a = 3, b = 2, and c = 5.

\frac{x^2}{9} + \frac{y^2}{4} = \frac{z^2}{25}

Substitute z = 5: Plug z = 5 into the equation and simplify the right side.

\frac{x^2}{9} + \frac{y^2}{4} = \frac{25}{25} = 1

Read off the ellipse: The cross section is the ellipse x²/9 + y²/4 = 1, which has semi-axes 3 (in x) and 2 (in y).

Answer: At height z = 5, the cross section is an ellipse with semi-axes of length 3 and 2.

Why It Matters

Elliptic cones appear in multivariable calculus and engineering when modeling structures or light beams that spread unevenly. Understanding them also builds intuition for quadric surfaces, a key topic in analytic geometry and linear algebra courses.

Common Mistakes

Mistake: Assuming every cone has circular cross sections.

Correction: A cone is circular only when a = b. When the two denominators under x² and y² differ, the cross sections are ellipses, making it an elliptic cone.