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 hyperboloid of one sheet.

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

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 parametric equations :

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

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

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

The cross sections of a hyperboloid are hyperbolas when taken with respect to the x-z or y-z planes and circles when taken with respect to the x-y plane. This site provides good interactive animations that help illustrate this.

Engineering Uses

The construction of a 1 sheeted hyperboloid can be approximated by having two circles connected by many straight beams.

The design has become the standard for nuclear cooling towers because of its minimum usage of material, structural strength, and improved convective air flow.

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