quadratic surfaces

The common equation of all quadratic surfaces is

$$A{x}^2+B{y}^2+C{z}^2+2A^{\prime }yz+2B^{\prime }zx+2C^{\prime }xy+2A^{\prime \prime }x+2B^{\prime \prime }y+2C^{\prime \prime }z+D=0$$

where $A,B,C,A^{\prime },B^{\prime },C^{\prime },A^{\prime \prime },B^{\prime \prime },C^{\prime \prime },D$ are constants and at least one of the six first does not vanish.  The different non-degenerate kinds are as follows; we give also the simplest equation.

This classification is based on examining the signature (http://planetmath.org/SylvestersLaw) of the quadratic form

$$A{x}^2+B{y}^2+C{z}^2+2A^{\prime }yz+2B^{\prime }zx+2C^{\prime }xy$$

and the signature of the form

$$A{x}^2+B{y}^2+C{z}^2+2A^{\prime }yz+2B^{\prime }zx+2C^{\prime }xy+2A^{\prime \prime }xw+2B^{\prime \prime }yw+2C^{\prime \prime }zw+D{w}^2$$

Note that, because of the fact that the equation describes the same surface if we simultaneously change the signs of all the coefficients, we obtain the same type of surface if we change all the signs in both signatures.

Surfaces without midpoints (http://planetmath.org/Midpoint3):

plotA.png

a) Elliptic paraboloid,  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=2z$

Signatures: $[++0]$, $[+++-]$ (or $[--0]$, $[+---]$)

plotB.png

b) Hyperbolic paraboloid,  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2z$;  it is a doubly ruled surface.

Signatures: $[+-0]$, $[++--]$

plotC.png

c) Parabolic cylinder,  $x^2=2pz$;  it is a developable surface.

Signatures: $[+00]$, $[++-0]$ (or $[-00]$, $[+--0]$)

Surfaces with one midpoint:

plotD

a) Ellipsoid,  $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

Signature: $[+++]$, $[+++-]$ (or $[---]$, $[+---]$)

plotE

b) One-sheeted hyperboloid,  $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$;  it is a doubly ruled surface.

Signatures: $[++-]$, $[++--]$ (or $[+--]$, $[++--]$)

plotF

c) Two-sheeted hyperboloid,  $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

Signature: $[++-]$, $[+++-]$ (or $[+--]$, $[+---]$)

plotG

d) ,  $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$;  it is a developable surface.

Signatures: $[++-]$, $[++-0]$ (or $[+--]$, $[+--0]$)

Surfaces with infinitely many midpoints

plotH

a) Hyperbolic cylinder,  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$;  it is a developable surface.

Signatures: $[+-0]$, $[+--0]$ (or $[+-0]$, $[++-0]$)

plotI

c) Elliptic cylinder,  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$;  it is a developable surface.

Signatures: $[++0]$, $[++-0]$ (or $[--0]$, $[+--0]$)

b) Two intersecting planes,  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$
Signatures: $[+-0]$, $[+-00]$

d) Two parallel planes,  $x^2=a^2$
Signatures: $[+00]$, $[+-00]$ (or $[-00]$, $[+-00]$)

e) Double plane,  $x^2=0$
Signatures: $[+00]$, $[+000]$ (or $[-00]$, $[-000]$)

Algebraically, there are other possibilities for the signatures, such as $[+++]$ and $[++++]$.  However, these give rise to equations which have no real solutions, hence they have been ignored.

Title	quadratic surfaces
Canonical name	QuadraticSurfaces
Date of creation	2013-03-22 14:59:40
Last modified on	2013-03-22 14:59:40
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	53
Author	pahio (2872)
Entry type	Topic
Classification	msc 51N20
Synonym	surfaces of second degree
Related topic	TangentPlaneOfQuadraticSurface
Related topic	Ellipsoid
Related topic	SurfaceOfRevolution2
Related topic	GeneratricesOfOneSheetedHyperboloid
Related topic	GeneratricesOfHyperbolicParaboloid
Related topic	AnalyticGeometry
Related topic	IntersectionOfQuadraticSurfaceAndPlane
Defines	elliptic paraboloid
Defines	hyperbolic paraboloid
Defines	parabolic cylinder
Defines	ellipsoid
Defines	one-sheeted hyperboloid
Defines	two-sheeted hyperboloid
Defines	cone surface
Defines	hyperbolic cylinder
Defines	elliptic cylinder