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Conic sections

$$Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F =0$$

can be classified as an ellipse, parabola, or hyperbola by looking at the determinant of a discriminant or as lines in the case the conic section is degenerate(see https://en.wikipedia.org/wiki/Degenerate_conic).

Can I similarly easily classify the angle of the axis of the hyperbola?

Can I impose that the axis should be horizontal with simple constraints on $A$,$B$,$C$,$D$,$E$ and $F$?

Can I do the same for a general angle?

Kvothe
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  • See, for instance, this answer, which gives formulas for many geometric elements of a general conic in terms of the coefficients of its equation. (The answer explicitly mentions ellipses, but the formulas can be adapted to hyperbolas as well.) – Blue Jan 25 '23 at 17:42
  • HINT. The eigenvectors of matrix $$\pmatrix{A & B\ B & C\}$$ give the directions of the axes. – Intelligenti pauca Jan 25 '23 at 20:53
  • @Intelligentipauca, thanks. How should I see which axis is which? From the sign of the related eigenvalue? – Kvothe Jan 26 '23 at 13:29
  • @Blue, such a list seems very useful if it were complete for other conic sections such as a hyperbola as well. I am sure the formula's can be adapted but I would not know how to. – Kvothe Jan 26 '23 at 13:31
  • @Kvothe: Some formulas "just work". Others need only a sign adjustment. Some might be trickier. (I haven't scrutinized them all.) That said, the presentation is somewhat (unnecessarily) complicated, so it can be difficult to determine what's what. Unfortunately, I don't know an alternative source off-hand. (Someday, I'll post my own clear and comprehensive answer to that question ... but I've been saying that for years. :) Good luck! – Blue Jan 26 '23 at 13:55
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    Answering my own question "The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis." https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections – Kvothe Jan 26 '23 at 14:12

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In Wikipedia's "Matrix representation of conic sections" entry we find that

[T]he two eigenvectors of the matrix of the quadratic form of a central conic section (ellipse or hyperbola) are perpendicular (orthogonal to each other) and each is parallel to (in the same direction as) either the major or minor axis of the conic. The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis.

Note that the major axis is usually called the transverse axis for a hyperbola and the minor axis the conjugate axis.

This makes it clear that to impose that the axes are vertical and horizontal one has to impose that $B=0$ and to impose specifically that the conjugate axis is horizontal one needs $A>C$. One also has to of course still impose that $A\cdot C < 0$ in order to have a hyperbola and not an ellipse.

(This leaves open to a generalization on how to impose any angle. A result which should follow by a simple rotation of the coordinates but I have not worked this out.)

Blue
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Kvothe
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