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Let's say we have a conic of the form

$$ax^2 + bxy + cy^2 + dx + ey + f = 0$$

How can I parameterize it? For ellipses there is a great answer. But how can I extend it for parabolas and hyperbolas as well? I understand I will need different parametric functions and also ranges.

Jean Marie
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vidstige
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  • Have you heard about the polar equation when center is one of its foci: $r=\frac{r_0}{1+e \cos(\theta-\theta_0)}$ common to all conics with $e$ the eccentricity ($0<e<1$ for an ellipse, $e=1$ for a parabola, $e>1$ for a hyperbola) ? It is one of the simplest representations. – Jean Marie Dec 19 '22 at 21:53
  • I've never heard about that. But it's not on parametric form, e.g. x(t) = ..., y(t) = ... – vidstige Dec 20 '22 at 10:11
  • There are many different ways to describe a curve : implicit $f(x,y)=0$, explicit $y=f(x)$, parametric $x(t),y(t)$, polar $r=r(\theta)$, etc... I am surprized you haven't yet encountered the last one which is very important and provides for curves like conic curves an especially easy unified treatment. – Jean Marie Dec 20 '22 at 10:54

1 Answers1

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The idea of parametrization in Parametrization of an arbitrary conic/ellipse can be further extended with a general, complex parameters. For example, the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ can be parametrized with $x=a\cos t$ and $y=b\sin t$. Very similarly, for the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, we can have $x=a\cos it$ and $y=bi\sin it$, where $i=\sqrt{-1}$.

P.S.

Through a combination of transportation and rotation, the general conic curve $ax^2+bxy+cy^2+dx+ey+f=0$ can be converted to $$ { Ax^2+By^2=C\quad,\quad \text{if $b^2-4ac\ne0$} \\ Ax^2+By=C\quad,\quad \text{if $b^2-4ac=0$} } $$

Mostafa Ayaz
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  • Ah, yes. I know it's possible. The question is how does the parameters of the hyperbolas, etc related to the original coefficients $a, b, c, d, e, f$? – vidstige Dec 20 '22 at 10:59