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# Euclidean Normal Forms of Three-Dimensional Quadrics

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

 normal form name point (double) cone line intersecting planes coincident planes

 normal form name (empty set) hyperboloid of 2 sheets hyperboloid of 1 sheet ellipsoid (empty set) hyperbolic cylinder elliptic cylinder (empty set) parallel planes

 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