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The general form of an ellipse is: $$A x^2 +B x y + C y^2 + D x + E y + F = 0$$ provided that $B^2 – 4 A C <0$

As an example, the equation below: $$-0.25 x^2 – 0.4 x y – 0.25 y^2 -1.5 x - 2 y + 1.5 = 0$$ is an ellipse, which plot is included below: enter image description here On the other way, he Wikipedia page, https://en.wikipedia.org/wiki/Ellipse, gives formulas (in paragraph “General ellipse”) that make connection between canonical form parameters (semi-axes, rotation angle, center coordinates) and general form coefficients (A,B,C,D,E,F).

These formulas imply in particular that $A$ and $C$ must be positive (as they are calculated adding squared values): $$A = a^2 \sin^2 \theta + b^2 \cos^2 \theta$$ $$ ...$$

As obviously (as shown in plot above), $A$ and $C$ do not need this condition to lead to draw an ellipse, there must be some sort of inconsistency in the definition of the general form of an ellipse…

Or there is something that escapes me in the way the formulas given by Wikipedia are obtained, or in the conditions under which these formulas can be obtained...

Has anyone ever asked this kind of question?

José Carlos Santos
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Andrew
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1 Answers1

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Note that$$AX^2+BXY+CY^2+DX+EY+F=0$$is equivalent to$$-AX^2-BXY-CY^2-DX-EY-F=0.$$On the other hand,\begin{align}B^2-4AC<0\implies &AC>0\\\iff&A\text{ and }C \text{ are both positive or both negative.}\end{align}So, if they are both negative you flip all the signs, and then you get the same ellipse, but now the coefficients of $X^2$ and $Y^2$ are positive.

José Carlos Santos
  • 427,504
  • Of course, that seems the simplest and only way to do... But I was wondering why the condition $B^2-4AC<0$ is always indicated (rightly so), but not the obligation to have $A>0$ and $B>0$, which appear to be necessary to make the link between general form parameters and canonical form parameters... I mean that, when you are given the "general" form, it seems that $A,B,...$ could take any real value when in fact, to be logical, we should indicate some restrictions on the signs of $A$ and $C$, right? – Andrew Jun 28 '23 at 08:24
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    Please bear in mind that the title of that section is “General ellipse” rather than “General formula for an ellipse”. The statement that every ellipse is of the form given at the article is correct. You seem to be aiming at every possible quadric that corresponds to an ellipse. If that's what you want then, indeed, the list mentioned in that article is incomplete, since it misses, so to say, half of them. – José Carlos Santos Jun 28 '23 at 08:49