the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the normal form of the equation, since two quadrics have the same normal form if and only if there is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows:
:
:
:
:where the are either 1, -1 or 0, except which takes only the value 0 or 1.

Each of these 17 normal forms