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# Euclidean Normal Forms of Two-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 intersecting pair of lines coincident lines

 normal form name (empty set) hyperbola ellipse (empty set) parallel pair of lines

 normal form name parabola

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.

 intersecting pair of lines coincident lines

 hyperbola ellipse

 parallel pair of lines parabola