Proof of Conics in R^2

**shadow_2145** · September 18th 2008, 04:00 PM

I really suck at proofs and I have this due tomorrow...any help would be appreciated. It doesn't count toward any points but I need to turn it in for credit

Definition: (1.3) Conics in $\mathbb{R}^2$ . A conic in $\mathbb{R}^2$ is a plane curve given by the quadratic equation $q(x,y) = ax^2 + bxy +cy^2 + dx +ey +f = 0$

Here is my question:
Prove the statement in (1.3) that an affine transformation can be used to put any conic of $\mathbb{R}^2$ into one of the standard forms (a-1). (Hint: use a linear transformation $x \to Ax$ to take the leading term $ax^2 + by + cy^2$ into one of $\underline{+}x^2\underline{+}y^2$ or $\underline{+}x^2$ or $0$ ; then complete the square in $x$ and $y$ to get rid of as much of the linear part as possible.

Thank you for your time and consideration. I will be up for awhile tonight to answer any questions and try to figure this out!

**kalagota** · September 19th 2008, 05:59 AM

Originally Posted by shadow_2145

I really suck at proofs and I have this due tomorrow...any help would be appreciated. It doesn't count toward any points but I need to turn it in for credit

Definition: (1.3) Conics in $\mathbb{R}^2$ . A conic in $\mathbb{R}^2$ is a plane curve given by the quadratic equation $q(x,y) = ax^2 + bxy +cy^2 + dx +ey +f = 0$

Here is my question:
Prove the statement in (1.3) that an affine transformation can be used to put any conic of $\mathbb{R}^2$ into one of the standard forms (a-1). (Hint: use a linear transformation $x \to Ax$ to take the leading term $ax^2 + by + cy^2$ into one of $\underline{+}x^2\underline{+}y^2$ or $\underline{+}x^2$ or $0$ ; then complete the square in $x$ and $y$ to get rid of as much of the linear part as possible.

Thank you for your time and consideration. I will be up for awhile tonight to answer any questions and try to figure this out!

well, i'll do this in the projective plane, that is you can always have a transformation from the affine plane to the projective plane.. (i'll just copy the notes i have here)

let $F(x,y,z) = ax^2 + bxy +cy^2 + dxz +eyz +fz^2$

case1: $a=c=f=0$
at least one of $b,d,e$ is not 0.
WLOG, let $b\not= 0$ .
consider the transformation:
$x' = x$
$y' = -x+y$
$z'=z$

arrive at:...

case2: at least one of $a,c,f$ is not 0.
WLOG, let $a\not=0$
and another WLOG, let $a=1$ .
$F(x,y,z) = x^2 + bxy +cy^2 + dxz +eyz +fz^2$
$= \left(x +\frac{b}{2}y+\frac{d}{2}z\right)^2 + \left(c-\frac{b^2}{4}\right)y^2 + \left(e-\frac{bd}{2}\right)yz + \left(f-\frac{d^2}{4}\right)z^2$

consider:
$x'=x + \frac{b}{2}y + \frac{d}{2}z \Rightarrow x = x' -\frac{b}{2}y'-\frac{d}{2}z'$
$y'=y$
$z'=z$

arrive at: ...
WLOG, assume $b=0,d=0$
left with: $F(x,y,z) = x^2 +cy^2+eyz+fz^2$

case3: from $F(x,y,z) = x^2 +cy^2+eyz+fz^2$
if $c=e=f=0$
$F(x,y,z)=x^2$

case4: from $F(x,y,z) = x^2 +cy^2+eyz+fz^2$ , at least one of $c,e,f$ is not = 0
if $e\not=0$ and $c=0,f=0$
$F(x,y,z)=x^2 + eyz$

consider:
$x'=x$
$y'=y-z$
$z'=z$

arrive at:
$F(x',y',z') = x'^2 + e(y'+z')z' = x'^2 + ey'z' + ez'^2$ or
$F(x,y,z) = x^2 + eyz + ez^2$

If $c\not=0$ , by interchanging $y$ and $z$ , we can force $f\not=0$ .

now, WLOG, let $f\not=0$

case5: $F(x,y,z) = x^2 +cy^2+eyz+fz^2$ , $f\not=0$

let $t = \sqrt{|f|}$ , $t^2 = \pm f$

consider $\varphi$ :
$x'=x$
$y'=y$
$z'=tz$

by this transformation, you shall arrive at:
$F^{\varphi}(x,y,z) = x^2 + \left(c \mp \frac{e^2}{4t^2}\right)y^2 \pm \left(\frac{e}{2t}y+z\right)^2$

consider $\psi$ :
$x'=x$
$y'=y$
$z'=\frac{e}{2t}y+z$

arrive at:
$(F^{\varphi})^{\psi}=F^{\psi\varphi} = x^2 + \left(c \mp\frac{e^2}{4t^2}\right)y^2 \pm z^2$

case6:
$F(x,y,z) = x^2 + Cy^2 \pm z^2$ , $C\not=0$

set $s = \sqrt{|C|}$
consider:
$x'=x$
$y'=sy$
$z'=z$

you arrive at: $F(x,y,z) =x^2 \pm y^2 \pm z^2$

so from there, consider all the combinations of plus and minus there..

case7: if $C=0$

$F(x,y,z)= x^2 \pm z^2$

consider:
$x'=x$
$y'=z$
$z'=y$

you arrive at: $F(x',y',z')=x'^2 \pm y'^2$ or simply $F(x,y,z)=x^2 \pm y^2$

-------------------end-------------------

well, the discussion is too long so try to understand it by yourself.. and this is just a guide or outline of the whole proof..

and another: the reason why there are 7 cases is that, this actually shows that any second degree polynomial in the affine plane can be transformed into one of the following:
1. empty set
2. single point
3. a doubled line i.e. two lines which are coinciding
4. two distinct lines
5. a conic..

you shall see which cases gives you the conics (which you are looking for)..

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