The Hyperboloid

Click for VRML

Introduction:

The hyperboloid is a well-known quadratic surface that comes in two varieties: the hyperboloid of one sheet (above) and the hyperboloid of two sheets (below).

They are so named because they consist of one and two connected pieces, respectively. It is interesting to note that the hyperboloid of one sheet is asymptotic to a cone, as shown below.

The hyperboloid of one sheet is also a ruled surface. That is, it contains at least one family of 1-parameter straight lines. The hyperboloid is reparameterized below to show this ruling more clearly:

Definition:

A hyperboloid of one sheet is parameterized by:

Implicitly, it is defined by:

The hyperboloid of two sheets is parameterized by:

It can also be defined implicitly as:

Properties:

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

H[u, v] = [click for formula]

Mean curvature of the surface.
Surface colored by Mean curvature.