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I refer to the Wikipedia page, https://en.wikipedia.org/wiki/Ellipse, giving formulas (in paragraph “General ellipse”) that make connection between canonical form parameters (semi-axes $a$ and $b$, rotation angle $\theta$, center coordinates $x_c$ and $y_c$) and general form coefficients ($A,B,C,D,E,F$).

Precisely, I am focusing on the equations: $$A = a^2 \sin^2 \theta + b^2 \cos^2 \theta$$ $$C = a^2 \cos^2 \theta + b^2 \sin^2 \theta$$ $$F = A x_c^2 + B x_c y_c + C y_c^2 - a^2 b^2$$

From which I can infer a system of $2$ equations with unknown variables $a^2$ and $b^2$: $$a^2 + b^2 = A + C$$ $$a^2 b^2 = A x_c^2 + B x_c y_c + C y_c^2 - F$$

Now, I introduce the following ellipse: $$0.2 x^2 + 0.4 x y + 0.25 y^2 + 1.5 x + 2 y - 1.5 = 0$$ $$(A=0.2, B=0.4, C=0.25, D=1.5, E=2, F=-1.5)$$

which center is given by the other $2$ equations from the same Wikipedia page (if I don't err): $$x_c = \frac{2CD - BE}{B^2 - 4AC} = 1.25$$ $$y_c = \frac{2AE - BD}{B^2 - 4AC} = -5$$

So that I am left with the equalities (if I didn't make any miscalculations...): $$a^2 + b^2 = 0.45$$ $$a^2 b^2 = 5.5625$$

I wonder how one can find $a^2$ and $b^2$ from these equalities...

Question in summary: what is the problem? where am I wrong?

Viktor Vaughn
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Andrew
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  • Numbers $A$, $B$ and $C$ in the formula for $F$ are not the coefficients in your equation for the conic, but rather the numbers obtained from the first three formulas on that Wikipedia page. And the results obtained by those formulas differ (in general) from the coefficients by an overall factor. – Intelligenti pauca Jun 28 '23 at 16:07
  • The equation can also be written as $$0.2 (x-1.25)^2 + 0.4 (x-1.25) (y+5)+ 0.25 (y+5)^2 - 5.5625=0.$$ To compare with $$((x-1.25)\cos{\theta}+(y+5)\sin{\theta})^2/a^2+(-(x-1.25)\sin{\theta}+(y+5)\cos{\theta})^2/b^2-1=0$$ do you see the need to scale by $5.5625$? Anyway $$a =\frac14 \sqrt{\frac{445}2 (9 + \sqrt{65})}\approx 15.6,$$

    $$b = \frac14 \sqrt{\frac{445}2 (9 - \sqrt{65})}\approx 3.7,$$

    $$\theta \approx 46.6^\circ$$

     – Jan-Magnus Økland Jun 29 '23 at 04:51
  • Please refer to my older post with numerical example here. – Ng Chung Tak Jun 29 '23 at 06:08
  • Thanks for detailing the problem and for the numerical example. It's now much clearer. I regret that the Wikipedia page was not explicit enough on how to interpret the parameters that are used... – Andrew Jun 29 '23 at 08:41

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Coefficients $A$, $B$, ... are defined up to a global multiplicative factor $k$. Hence the right equations should be:

$$kA = a^2 \sin^2 \theta + b^2 \cos^2 \theta$$ $$kC = a^2 \cos^2 \theta + b^2 \sin^2 \theta$$

and so on, where $k$ is unknown.

Intelligenti pauca
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  • I'm a little disturbed by the introduction of this factor k... and I don't quite understand what the next steps are. Finally, with your method, what system of two equations in the unknowns $a^2$ and $b^2$ do you arrive at? – Andrew Jun 28 '23 at 19:29
  • @Andrew On that Wikipedia page you can also find the reverse formulas: is there anything wrong with them? – Intelligenti pauca Jun 28 '23 at 21:01
  • Yes! The computation of $a$, $b$ is not explained. The page indicates "citation needed"... That's why I found it would be useful to derive myself the values of $a$ and $b$ by manipulating the equations written a little above in the same page. These equations give us $a^2$ and $b^2$ with logically no problem. But you saw (in my example given) that this looks apparently not possible. Why? I just used the given equations... What is the mistake? What is the fault? How to do it in practice? – Andrew Jun 28 '23 at 21:31
  • It's written there: those formulas for $A$, $B$ etc. will give you A PARTICULAR set of coefficients if you know $a$, $b$ etc. To reverse them you must introduce a scale factor as I explained above. – Intelligenti pauca Jun 28 '23 at 21:36