Mathematics-Online lexicon: Euclidean Normal Forms of Three-Dimensional Quadrics

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	Mathematics-Online lexicon:
	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

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+\frac{x_3^2}{a_3^2}=0$ point

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}-\frac{x_3^2}{a_3^2}=0$ (double) cone

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}=0$ line

$\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}=0$ intersecting planes

$\frac{x_1^2}{a_1^2}=0$ coincident planes

central quadrics

normal form name

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+\frac{x_3^2}{a_3^2}+1=0$ (empty set)

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}-\frac{x_3^2}{a_3^2}+1=0$ hyperboloid of 2 sheets

$\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}-\frac{x_3^2}{a_3^2}+1=0$ hyperboloid of 1 sheet

$-\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}-\frac{x_3^2}{a_3^2}+1=0$ ellipsoid

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+1=0$ (empty set)

$\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}+1=0$ hyperbolic cylinder

$-\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}+1=0$ elliptic cylinder

$\frac{x_1^2}{a_1^2}+1=0$ (empty set)

$-\frac{x_1^2}{a_1^2}+1=0$ parallel planes

parabolic quadrics

normal form name

$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+2x_3=0$ elliptic paraboloid

$\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}+2x_3=0$ hyperbolic paraboloid

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

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

(double) cone intersecting planes

$\includegraphics[width=.4\moimagesize]{quadriken_kegel}$ $\includegraphics[width=.4\moimagesize]{quadriken_schneidende_ebenen}$

hyperboloid of 2 sheets hyperboloid of 1 sheet

$\includegraphics[width=.4\moimagesize]{quadriken_zweischaliges_hyperboloid}$ $\includegraphics[width=.4\moimagesize]{quadriken_einschaliges_hyperboloid}$

ellipsoid hyperbolic cylinder

$\includegraphics[width=.4\moimagesize]{quadriken_ellipsoid}$ $\includegraphics[width=.4\moimagesize]{quadriken_hyperbolischer_zylinder}$

elliptic cylinder elliptic paraboloid

$\includegraphics[width=.4\moimagesize]{quadriken_elliptischer_zylinder}$ $\includegraphics[width=.4\moimagesize]{quadriken_paraboloid}$

hyperbolic paraboloid parabolic cylinder

$\includegraphics[width=.4\moimagesize]{quadriken_hyperbolisches_paraboloid}$ $\includegraphics[width=.4\moimagesize]{quadriken_parabolischer_zylinder}$

see also:

Keyword: Quadric

[Examples]

automatically generated 7/13/2018

normal form	name
$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+\frac{x_3^2}{a_3^2}=0$	point
$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}-\frac{x_3^2}{a_3^2}=0$	(double) cone
$\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}=0$	line
$\frac{x_1^2}{a_1^2}-\frac{x_2^2}{a_2^2}=0$	intersecting planes
$\frac{x_1^2}{a_1^2}=0$	coincident planes

(double) cone	intersecting planes

$\includegraphics[width=.4\moimagesize]{quadriken_kegel}$	$\includegraphics[width=.4\moimagesize]{quadriken_schneidende_ebenen}$

hyperboloid of 2 sheets	hyperboloid of 1 sheet

$\includegraphics[width=.4\moimagesize]{quadriken_zweischaliges_hyperboloid}$	$\includegraphics[width=.4\moimagesize]{quadriken_einschaliges_hyperboloid}$

ellipsoid	hyperbolic cylinder

$\includegraphics[width=.4\moimagesize]{quadriken_ellipsoid}$	$\includegraphics[width=.4\moimagesize]{quadriken_hyperbolischer_zylinder}$

elliptic cylinder	elliptic paraboloid

$\includegraphics[width=.4\moimagesize]{quadriken_elliptischer_zylinder}$	$\includegraphics[width=.4\moimagesize]{quadriken_paraboloid}$

hyperbolic paraboloid	parabolic cylinder

$\includegraphics[width=.4\moimagesize]{quadriken_hyperbolisches_paraboloid}$	$\includegraphics[width=.4\moimagesize]{quadriken_parabolischer_zylinder}$