CTK Insights

A hyperbola has two axes of symmetry: one that crosses the hyperbola while the other does not. Two different 3D shapes are obtained when a hyperbola is made to rotate around its axes.

Two sheet hyperboloid	One sheet hyperboloid

The equation of one is

$-\frac{x^2}{a^2}-\frac{y^2}{a^2}+\frac{z^2}{b^2}=1.$

and that of the other is

$\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{b^2}=1.$

The one sheet hyperboloid has a rather unexpected property. It's surface can be entirely covered by straight lines. In fact, there are two families of straight lines that cover a one sheet hyperboloid.

This tells us that the one sheet hyperboloid can be constructed starting with a cylinder and then rotating one of its bases relative to the other.

Here's a picture of a half empty toothpick container in which the toothpicks have spontaneously arranged to form the hyperboloid:

(There is a dynamic illustration.)

Related posts:

1. Golden Ratio - Another Sighting