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Can i treat a hyperbola as a special case of ellipse. Like substituting $b^2$ with $-b^2$. Would all things still work?

And also, why is a parabola different from the family of (circle, ellipse, hyperbola)?

Or am I not looking at it correctly? Thanks!

Ng Chung Tak
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Adam Karlson
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    Depends on what you mean with different. Parabolas are no more different than circles, see https://en.wikipedia.org/wiki/Conic_section#Discriminant_classification – gammatester Jan 13 '16 at 11:16
  • They are all the same in projective geometry. But in affine geometry, including ordinary Euclidean geometry, they are different. You can see this because they have such different shapes. For example, a hyperbola is in two pieces while an ellipse is in one. Parabolas are different from ellipses because they are unbounded. And they're different from hyperbolas because they don't have asymptotes. Parabolas also don't have a centre of symmetry, unlike all the other conics. – David Jan 13 '16 at 11:23
  • Is it due to a lack of (or existence of) a centre of symmetry, that parabola does not have angular parametric coordinates (while others have)? – Adam Karlson Jan 14 '16 at 13:27
  • "Would all things still work": this is a too broad statement, narrow it down. What are you thinking of ? –  Jul 29 '16 at 13:42
  • What do you mean by angular parametric coordinates? Are you talking about polar coordinates, like how a circle can be described by $r + 4\cos\theta = 0$, for example? –  Jul 29 '16 at 13:43

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If you agree to work in the (real or complex) projective plane, the ellipse, the hyperbola and the parabola are essentially the same thing: consider, for simplicity, the ellipse $x^2 + y^2 = 1$ (a circle). Projectivize it, by replacing $x$ and $y$ by $\frac x z$ and $\frac y z$, with $z \ne 0$. You will get $x^2 + y^2 - z^2 = 0$. Good.

Think now of the hyperbola $x^2 - y^2 = 1$. Projectivizing it like above, you get $z^2 + y^2 = x^2$, so by a change the coordinates $(X,Y,Z) = (z,y,x)$ this becomes $X^2 + Y^2 - Z^2 = 0$, precisely like the ellipse above.

Now, let's take a look at the parabola $y = x^2$. Its projectivized equation is $zy = x^2$. Consider the change of variables $X = x, \ Y = \frac {y-z} 2, \ Z = \frac {y+z} 2$. Then the equation becomes $(Y+Z) (Z-Y) = X^2$, which again is just $X^2 + Y^2 - Z^2 = 0$, again like the ellipse.

If you don't want to work in the projective plane, but are at least willing to switch to complex coordinates, then the ellipse $x^2 + y^2 = 1$ is readily transformed into $X^2 - Y^2 = 1$ by the change of variables $x = X, \ y = \Bbb i Y$. You can't do this with the parabola, though. In the non-projective world, the parabola belongs to a different species that the ellipse and the hyperbola.

If you want to work neither with projective coordinates, nor with complex ones, then the three curves that you mention are completely different and there is no way to reconcile them.

The arguments illustrated above can be made to work on ellipses, hyperbolae and parabolae given by arbitrary equations, not just by the simple ones discussed, but I feel that for the purpouse of illustrating the underlying idea the chosen ones are just enough.

Alex M.
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There is a unifying definition of the conics, based on a directrix line and a focus point, expressing that the ratio of the distance to the directrix andto the focus is a constant.

https://en.wikipedia.org/wiki/Conic_section#Eccentricity.2C_focus_and_directrix

When you increase the eccentricity, the conic which is first an ellipse starts growing and its center moves away from the directrix; at some point it goes to infinity, turning the ellipse to a parabola; then the center comes back from infinity on the other side while the curve changes to a hyperbola.

This is reflected in the equation of the conic in polar coordinates

$$\rho=\frac{l}{1+e\cos\theta}.$$

For $e<1$, the denominator doesn't cancel and this yields a closed curve, i.e. an ellipse. For $e>1$ there are two roots, corresponding to the two asymptotes of an hyperbola. And of course in between a single root yields the parabola.

In this representation, the three types of curves form a continuum. On the opposite, if you look at the centered conics, the parabola looks different as it "center" would be at infinity.