# Difference between revisions of "Hyperboloid"

Hyperboloid
Field: Calculus
Image Created By: Paul Nylander
Website: Paul Nylander's Site

Hyperboloid

A hyperboloid is a quadric, a type of surface in three dimensions.

# Basic Description

A hyperboloid is a quadric surface in 3 dimensions that is generated by rotating a hyperbolic function around an axis.

The hyperbolic function $sin{hx} \,$ produces the graph on the left, and by rotating it about the y-axis we get a hyperboloids of one sheet.

The hyperbolic function $coshx\,$ produces the graph on the left, and by rotating it about the y-axis we get a hyperboloids of one sheet.

They are represented by the following equations.

${x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over c^2}=1 \,$  hyperboloid of one sheet,

or

$- {x^2 \over a^2} - {y^2 \over a^2} + {z^2 \over c^2}=1 \,$  hyperboloid of two sheets.

These curves can also be defined using the following

$x(u,v)=a(\sqrt{1+u^2})cos{v}\,$

$y(u,v)=a(\sqrt{1+u^2})sin{v}\,$

$z(u,v)=cu\,$

## Analyzing a Hyperboloid from 2 Dimensionals

The cross sections of a hyperboloid are . This site provides good interactive animations that help illustrate this.

## Other Uses

The construction of a 1 sheeted hyperboloid can be approximated by having two circles connected by many straight beams. This structure is used most recognizably in nuclear cooling towers.