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I'm trying to prove that, if $C$ is a conic, then there exists a bijective affine transformation (BAT) that maps $C$ to a special instance, and conversely. Specifically,

  • a) $C$ is an ellipse iff there exists a BAT from $C$ to the unit circle $x^2+y^2=1$.
  • b) $C$ is an hyperbola iff there exists a BAT from $C$ to the hyperbola $x^2 - y^2 =1$.
  • c) $C$ is a parabola iff there exists a BAT from $C$ to the parabola $y^2=x$.

Deduce from these statements, that, given two non-degenerate conics $C$, $C^\prime$ there exists a BAT $f$ so that $f(C)=C^\prime$ iff both conics are ellipses, both are hyperbolas, or both are parabolas.

The types of conics are defined as:

  • An ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

  • A hyperbola is a set of points, such that for any point $P$ of the set, the absolute difference of the distances $|PF_1|$, $|PF_2|$ to two fixed points $F_1$, $F_2$ (the foci), is constant, usually denoted by $2a$, where $a>0$.

  • A parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus.


About a), b) and c) "ifs" I have the following:

For some orthonormal reference system where major and minor axes are aligned with the x,y-axes and the center of the elipse is $(0,0)$, its equation is $ (\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$. Then by the transformation $(x',y')=T(x,y)=(\frac{x}{a},\frac{y}{b})$ it is mapped to the unit circle.

I use the same transformation for the hyperbola.

For the parabola, given an orthonormal reference system its equation is $y^2=2px$, for $p>0$. So the transformation $T(x,y)=(\frac{x}{2p},y)$, I believe it to be suitable.

My issue is proving that if those affine transformations exist (in each case) then $C$ is an ellipse, a parabola, or a hyperbola. (The “only ifs”.)

Thanks so much in advance.

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  • $\begingroup$ To answer the question, one needs some starting definition of ellipse, hyperbola and parabola. $\endgroup$ – Yves Daoust Oct 23 '18 at 21:51
  • $\begingroup$ Hi! I felt the question was getting really long, and I decided to omit that information. I'll edit it with the definitions. $\endgroup$ – M. Navarro Oct 23 '18 at 21:56
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    $\begingroup$ See, imagine that I define an ellipse as "a curve that maps to a circle by an affine transform". $\endgroup$ – Yves Daoust Oct 23 '18 at 21:58

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