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# quadratic surfaces

The common equation of all quadratic surfaces is

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

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 of the quadratic form

$Ax^{2}+By^{2}+Cz^{2}+2A^{{\prime}}yz+2B^{{\prime}}zx+2C^{{\prime}}xy$ |

and the signature of the form

$Ax^{2}+By^{2}+Cz^{2}+2A^{{\prime}}yz+2B^{{\prime}}zx+2C^{{\prime}}xy+2A^{{% \prime\prime}}xw+2B^{{\prime\prime}}yw+2C^{{\prime\prime}}zw+Dw^{2}$ |

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 midpoints:

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

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

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

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

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

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

Surfaces with one midpoint:

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

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

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 $[+--]$, $[++--]$)

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

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

$\Delta<0$

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 midpoints

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

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

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 $[-00]$, $[-000]$)

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51N20*no label found*

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## Comments

## temporarily broken

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

## Re: temporarily broken

I'll try to get the picturesi n EPS as soon as possible, to make your entry work again

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: temporarily broken

The first picture is fine! I believe that fitting of such pictures is rather difficult.

Best regards,

Jussi

## Re: temporarily broken

I meant when yo uview the entry in html mode, the graphic for a)

shows besides b) (and I can't figure out why)

and the white background becomes black

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: temporarily broken

And, in page image mode, "setgray" stuff appears on the top of the page and the last line is cut off.

By the way, I actually like the black background. I think it looks really nice and makes the picture stand out better.

Also, this must be some kind of record on Planet Math --- the entry went up within a day of the request!! Usually these things take months. In fact, it went up so fast that there was no time to move it down to the fulfilled list. I'll move it there so you can erase it from the requests list.

Ray

## Re: temporarily broken

;)

all I know is that I had to upload like 10 differnet files o make the entry render due the tetex version pm uses has very old packages.

I think that somehow make it break the background, but hey, I also like better the black background

on the other hand, the picture that should go below a) still moves around (perhaps the \item is acting weird, I don't know)

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: temporarily broken

Hi, isn't the double plane a degenerate case? A straight line cannot intersect it in exactly two points.

## Re: temporarily broken

Of course it is a degenerate case. I only included it so as to be complete. Maybe it would be better to make a separate heading for degenerate cases and move it there.

Also, the case of two planes (which could be subdivided into the case of two parallel planes and case of two intersecting planes) is usually classified as a degenerate case in most of the algebraic geometry books I am used to because of the fact that, even though a generic line intersects it in two points, it can be expressed as the union of two first degree surfaces (planes). I suppose that there are different ways of defining degenerate. Maybe some algebraic geometers in the mebership could add their opinions on how they understand the word "degenrate". I know we have a lot of differential goemeters around here, but maybe not so many algebraic geometers.