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quadratic surfaces (Topic)

The common equation of all quadratic surfaces is

$\displaystyle Ax^2+By^2+Cz^2+2A'yz+2B'zx+2C'xy+2A”x+2B”y+2C”z+D = 0$
where $ A, B, C, A', B', C', A”, B”, C”, 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 of the quadratic form

$\displaystyle Ax^2+By^2+Cz^2+2A'yz+2B'zx+2C'xy$
and the signature of the form
$\displaystyle Ax^2+By^2+Cz^2+2A'yz+2B'zx+2C'xy+2A”xw+2B”yw+2C”zw+Dw^2$
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 centre:

\includegraphics{plotA}
a) Elliptic paraboloid, $ \frac{x^2}{a^2}+\frac{y^2}{b^2} = 2z$
Signatures: $ [++0]$, $ [+++-]$ (or $ [-0]$, $ [+--]$)


\includegraphics{plotB}
b) Hyperbolic paraboloid, $ \frac{x^2}{a^2}-\frac{y^2}{b^2} = 2z$; it is a doubly ruled surface.
Signatures: $ [+-0]$, $ [++-]$


\includegraphics{plotC}


c) Parabolic cylinder, $ x^2 = 2pz$; it is a developable surface.
Signatures: $ [+00]$, $ [++-0]$ (or $ [-00]$, $ [+-0]$)


Surfaces with one centre point:

\includegraphics{plotD}
a) Ellipsoid, $ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$
Signature: $ [+++]$, $ [+++-]$ (or $ [--]$, $ [+--]$)


\includegraphics{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 $ [+-]$, $ [++-]$)


\includegraphics{plotF}
c) Two-sheeted hyperboloid, $ \frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2} = 1$
Signature: $ [++-]$, $ [+++-]$ (or $ [+-]$, $ [+--]$)
$ \Delta < 0$


\includegraphics{plotG}
d) Cone surface, $ \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 centre points

\includegraphics{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]$)


\includegraphics{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.



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"quadratic surfaces" is owned by pahio. [ full author list (4) ]
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See Also: tangent plane of quadratic surface, ellipsoid, surface of revolution, generatrices of one-sheeted hyperboloid, generatrices of hyperbolic paraboloid, analytic geometry

Other names:  surfaces of second degree
Also defines:  elliptic paraboloid, hyperbolic paraboloid, parabolic cylinder, ellipsoid, one-sheeted hyperboloid, two-sheeted hyperboloid, cone surface, hyperbolic cylinder, elliptic cylinder
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Cross-references: solutions, real, parallel, planes, point, developable surface, doubly ruled, centre, type, coefficients, surface, quadratic form, signature, non-degenerate, vanish, equation
There are 10 references to this entry.

This is version 35 of quadratic surfaces, born on 2005-02-02, modified 2006-11-01.
Object id is 6700, canonical name is QuadraticSurfaces.
Accessed 13418 times total.

Classification:
AMS MSC: 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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temporarily broken by drini on 2005-02-02 10:30:56
I'm sorry I broke this entry, pahio, I wanted to add illustrations for the graphics, but it's behaving weirdly page iamge vs html mode
(image misplaced, colors wrong, no equations being displayed)

y the way, a possible cause for all these pstricks and graphics problems might be that tetex version installed on planetmath is waytoo old (and thus breaks things dealing with new xkeyval package), but that's just a conjecture
 f
G -----> H G
p \ /_ ----- ~ f(G)
 \ / f ker f
 G/ker f 
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