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Math Help - Proof of Conics in R^2

  1. #1
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    Proof of Conics in R^2

    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!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by shadow_2145 View Post
    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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