math lessons - Hyperboloid

Hyperboloid of one sheet

Hyperboloid of two sheets

In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:

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

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

If, and only if, $a = b$ , it is a hyperboloid of revolution. A hyperboloid of one sheet can be obtained by revolving a hyperbola around its transversal axis. Alternatively, a hyperboloid of two sheets of axis AB is obtained as the set of points P such that AP-BP is a constant, AP being the distance between A and P. A and B are then called the foci of the hyperboloid. A hyperboloid of two sheets can be obtained by revolving a hyperbola around its focal axis.

A hyperboloid of one sheet is a ruled surface; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.

A degenerate hyperboloid is of the form:

${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}=0;$