Cones

The Elliptic Cone

The Standard Equation

The Geometric Properties

• The vertex: at the origin $(0, 0, 0)$
• The generators: all the straight lines through the vertex
• Symmetry: symmetric about the three coordinate planes
• The direction it opens: it opens in both the positive and the negative $z$-directions

The Properties of Cones

The Nature of the Sections

The Behaviour at Infinity

A cone has the behaviour at infinity: as a point moves away from the vertex, the surface approaches infinity.

The Equation of a Generator

Every generator of the cone satisfies:

$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$

Special Cases

The Circular Cone

When $a = b$ in the elliptic cone, it becomes a circular cone:

$\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 0$

that is: $x^2 + y^2 = \frac{a^2}{c^2} z^2$

The Equilateral Cone

When $a = b = c$:

$x^2 + y^2 - z^2 = 0$

This is an equilateral elliptic cone.

Applications of Cones

Geometric Construction

In three-dimensional modelling, cones are used to create cone-shaped objects.

Optics

In optics, cones are used to describe the propagation paths of light rays.

Engineering Applications

In mechanical engineering, cones are used to design conical parts.

---

Exercises

Exercise 1

Determine the type of the surface $x^2 + y^2 - z^2 = 0$.

Exercise 2

Find an equation of one generator of the elliptic cone $\frac{x^2}{4} + \frac{y^2}{9} - \frac{z^2}{16} = 0$.

---

Summary

Symbols Used in This Article