1. ## Diagonal ellipse

Does anyone know a way to graph an ellipse such that the slope of the line segment between the foci is nonzero and also non-infinite? (Basically, does anyone know a Cartesian equation for a diagonal ellipse?) Parametric works.

2. ## Re: Diagonal ellipse

$ax^2 + bxy + cy^2 + dx + ey + f = 0$ is the equation for the general "rotated & shifted" conic section in the plane.

If $D = b^2- 4ac$, then it's an ellipse for $D<0$, a parabola for $D = 0$, and a hyperbola for $D>0$.

The a, b, c give the general shape & type (via D), and then the e and f give the translation. The bxy term is what appears when the conics are rotated from their usual positions nicely aligned to the x & y axes.

3. ## Re: Diagonal ellipse

Thank you so much! I will definitely print this out. Love you, stranger! (In a platonic way.)

4. ## Re: Diagonal ellipse

If you're going to print it out, I should point out a tiny error I now see in what I wrote. My original "e and f give the translation" was incorrect. It shoud read "e and d give the translation."

Also, that doesn't mean that you can read off the x and y translations from e and d - the relationship is more complicated. It only means that when those terms show up, it indicates that the conic section is no longer centered at the origin.

5. ## Re: Diagonal ellipse

The general formula for an ellipse with center at (a, b) and axes parallel to the x and y axes is $\frac{x^2}{a^2}+\frac{y^2}{b^2}= 1$ or $b^2x^2+ a^2y^2= a^2b^2$. To rotate so that the axis parallel to the x-axis moves to angle $\theta$ with the x-axis, substitute $cos(\theta)+ ysin(\theta)$ for x and $-y'sin(\theta)+ x'cos(\theta)$ for y.

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# how to calculate diagonal of selipse

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