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## Euclidean Normal Forms of Three-Dimensional Quadrics |

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z | overview |

There exist appropriate Cartesian coordinate systems with respect to which the equations defining quadrics have the following normal forms.

**conical quadrics**

normal form | name |

point | |

(double) cone | |

line | |

intersecting planes | |

coincident planes |

**central quadrics**

normal form | name |

(empty set) | |

hyperboloid of 2 sheets | |

hyperboloid of 1 sheet | |

ellipsoid | |

(empty set) | |

hyperbolic cylinder | |

elliptic cylinder | |

(empty set) | |

parallel planes |

**parabolic quadrics**

normal form | name |

elliptic paraboloid | |

hyperbolic paraboloid | |

parabolic cylinder |

The normal forms are uniquely determined up to permutation of subscripts and in the case of conical quadrics up to multiplication by a constant .

The values are set to be positive and are called lengths of the principal axes of the quadric.

(double) cone | intersecting planes |

hyperboloid of 2 sheets | hyperboloid of 1 sheet |

ellipsoid | hyperbolic cylinder |

elliptic cylinder | elliptic paraboloid |

hyperbolic paraboloid | parabolic cylinder |

automatically generated 7/13/2018 |