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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.


conical quadrics


normal form name
$ \frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}=0$ point
$ \frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}=0$ intersecting pair of lines
$ \frac{x_1^2}{a_1^2}=0$ coincident lines


central quadrics


normal form name
$ \frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+1=0$ (empty set)
$ \frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}+1=0$ hyperbola
$ -\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}+1=0$ ellipse
$ \frac{x_1^2}{a_1^2}+1=0$ (empty set)
$ -\frac{x_1^2}{a_1^2}+1=0$ parallel pair of lines


parabolic quadrics


normal form name
$ \frac{x_1^2}{a_1^2}+2x_2=0$ 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 $ c\ne 0$.

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

intersecting pair of lines coincident lines
\includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_5} \includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_6}

hyperbola ellipse
\includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_3} \includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_1}

parallel pair of lines parabola
\includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_4} \includegraphics[width=.4\moimagesize]{a_normalform_quadrik_2d_2}

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  automatically generated 7/13/2018