Editing Quadric (section)

== Euclidean space ==
In three-dimensional [[Euclidean space]], quadrics have dimension two, and are known as '''quadric surfaces'''. Their [[quadratic equation]]s have the form
:<math>A x^2 + B y^2 + C z^2 + D xy + E yz + F xz + G x + H y + I z + J = 0,</math>
where <math>A, B, \ldots, J</math> are real numbers, and at least one of {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} is nonzero.

The quadric surfaces are classified and named by their shape, which corresponds to the [[orbit (group theory)|orbits]] under [[affine transformation]]s. That is, if an affine transformation maps a quadric onto another one, they belong to the same class, and share the same name and many properties.

The [[principal axis theorem]] shows that for any (possibly reducible) quadric, a suitable change of [[Cartesian coordinates]] or, equivalently, a [[Euclidean transformation]] allows putting 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 [[Canonical form|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:
:<math> {x^2 \over a^2} + {y^2 \over b^2} +\varepsilon_1  {z^2 \over c^2} + \varepsilon_2=0,</math>
:<math> {x^2 \over a^2} - {y^2 \over b^2}  + \varepsilon_3=0</math>
:<math>{x^2 \over a^2}  + \varepsilon_4 =0,</math>
:<math>z={x^2 \over a^2} +\varepsilon_5 {y^2 \over b^2}, </math>
where the <math>\varepsilon_i</math> are either 1, −1 or 0, except <math> \varepsilon_3 </math> which takes only the value 0 or 1.

Each of these 17 normal forms<ref name="ela">Stewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996.</ref> corresponds to a single orbit under affine transformations. In three cases there are no real points: <math>\varepsilon_1=\varepsilon_2=1</math> (''imaginary ellipsoid''), <math>\varepsilon_1=0, \varepsilon_2=1</math> (''imaginary elliptic cylinder''), and <math>\varepsilon_4=1</math> (pair of [[complex conjugate]] parallel planes, a reducible quadric). In one case, the ''imaginary cone'', there is a single point (<math>\varepsilon_1=1, \varepsilon_2=0</math>). If <math>\varepsilon_1=\varepsilon_2=0,</math> one has a line (in fact two complex conjugate intersecting planes). For <math>\varepsilon_3=0,</math> one has two intersecting planes (reducible quadric).  For <math>\varepsilon_4=0,</math> one has a double plane. For <math>\varepsilon_4=-1,</math> one has two parallel planes (reducible quadric).

Thus, among the 17 normal forms, there are nine true quadrics: a cone, three cylinders (often called degenerate quadrics) and five non-degenerate quadrics ([[ellipsoid]], [[paraboloid]]s and [[hyperboloid]]s), which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid (no real point), the imaginary cylinder (no real point), the imaginary cone (a single real point), and the reducible quadrics, which are decomposed in two planes; there are five such decomposed quadrics, depending whether the planes are distinct or not, parallel or not, real or complex conjugate.

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Non-degenerate real quadric surfaces
|-
| &nbsp; &nbsp; [[Ellipsoid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,</math>
|[[Image:Ellipsoid Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Elliptic paraboloid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,</math>
|[[Image:Paraboloid Quadric.Png|150px]]
|-
| &nbsp; &nbsp; [[Hyperbolic paraboloid]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} - z = 0  \,</math>
|[[Image:Hyperbolic Paraboloid Quadric.png|150px]]
|-
| &nbsp; &nbsp;[[Hyperboloid of one sheet]] <br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Hyperbolic hyperboloid
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,</math>
|[[Image:Hyperboloid Of One Sheet Quadric.png|150px]]
|-
| &nbsp; &nbsp;[[Hyperboloid of two sheets]] <br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Elliptic hyperboloid
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,</math>
|[[Image:Hyperboloid Of Two Sheets Quadric.png|150px]]
|}

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Degenerate real quadric surfaces
|-
| &nbsp; &nbsp; [[Elliptic cone]]<br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Conical quadric
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,</math>
|[[Image:Elliptical Cone Quadric.Png|150px]]
|-
| &nbsp; &nbsp; [[Elliptic cylinder]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} = 1 \,</math>
|[[Image:Elliptic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Hyperbolic cylinder]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} = 1 \,</math>
|[[Image:Hyperbolic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp;  [[Parabolic cylinder]]
| <math>x^2 + 2ay = 0 \,</math>
|[[Image:Parabolic Cylinder Quadric.png|150px]]
|}

When two or more of the parameters of the canonical equation are equal, one obtains a quadric [[surface of revolution|of revolution]], which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere).

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Quadrics of revolution
|-
| &nbsp; &nbsp; Oblate and prolate [[spheroid]]s (special cases of ellipsoid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,</math>
|[[Image:Oblate Spheroid Quadric.png|75px]][[Image:Prolate Spheroid Quadric.png|75px]]
|-
| &nbsp; &nbsp; [[Sphere]] (special case of spheroid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,</math>
|[[Image:Sphere Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Circular paraboloid]] (special case of elliptic paraboloid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - z = 0  \,</math>
|[[Image:Circular Paraboloid Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Hyperboloid of revolution]] of one sheet (special case of hyperboloid of one sheet)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 1 \,</math>
|[[Image:Circular Hyperboloid Of One Sheet Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Hyperboloid of revolution]] of two sheets (special case of hyperboloid of two sheets)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = -1 \,</math>
|[[Image:Circular Hyperboloid of Two Sheets Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Circular cone]] (special case of elliptic cone)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 0 \,</math>
|[[Image:Circular Cone Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Circular cylinder]] (special case of elliptic cylinder)
| <math>{x^2 \over a^2} + {y^2 \over a^2} = 1  \,</math>
|[[Image:Circular Cylinder Quadric.png|150px]]
|}