A Quadratic Surface which may be one- or two-sheeted.

The one-sheeted circular hyperboloid
is a doubly Ruled Surface. When oriented along the *z*-Axis, the one-sheeted circular hyperboloid
has Cartesian Coordinates equation

(1) |

(2) | |||

(3) | |||

(4) |

for (left figure). Other parameterizations include

(5) | |||

(6) | |||

(7) |

(middle figure), or

(8) | |||

(9) | |||

(10) |

(right figure). An obvious generalization gives the one-sheeted Elliptic Hyperboloid.

A two-sheeted circular hyperboloid oriented along the *z*-Axis has Cartesian Coordinates equation

(11) |

(12) | |||

(13) | |||

(14) |

for . Note that the plus and minus signs in correspond to the upper and lower sheets. The two-sheeted circular hyperboloid oriented along the

(15) |

(16) | |||

(17) | |||

(18) |

(Gray 1993, p. 313). Again, an obvious generalization gives the two-sheeted Elliptic Hyperboloid.

The Support Function of the hyperboloid of one sheet

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

**References**

Fischer, G. (Ed.). Plates 67 and 69 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, pp. 62 and 64, 1986.

Gray, A. ``The Hyperboloid of Revolution.'' §18.5 in
*Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 296-297, 311-314, and 369-370, 1993.

© 1996-9

1999-05-25