Over the complex numbers, the above classification would become even simpler, because multiplication by $i$ from both sides will convert between $-1$ and $+1$. However there is also a class of quadratic forms which has no real solutions, and if you consider these as conics as well, then no, they are not equivalent to the former. Ellipses, hyperbolas and parabolas do fall into this category. ellipses, hyperbolas, and parabolas), still equivalent in $ℝℙ^2$, the same way they are equivalent in $ℂℙ^2$?Īll non-degenerate conics which do contain real points are projectively equivalent. A single real line with multiplicity 2.Īre all non-degenerate conic sections (i.e. A pair of conjugate complex lines, with their real point of intersection as the only real point in the conic. Purely complex despite the real coefficients. That's a non-degenerate conic which contains no real points. That's your regular real non-degenerate conic.
So at the end of the day, you can say that every real conic is up to projective transformations equivalent to one where the diagonal elements are one of the following: And you can also replace $A$ by $-A$ and still describe the same conic. Now you can do some permutations of coordinates, to rearrange elements. What is the definition of a projective change of coordinates in $\mathbb$. However, this change of coordinates is obviously not available in the real projective plane, so it is no longer obvious to me whether hyperbolas and ellipses are equivalent in the real projective plane.ġ. To transform between the unit ellipse (circle) $x^2 + y^2 =1$ and the unit hyperbola $x^2 - y^2=1$, we can use the simple change of coordinates $y_1 = iy_2$ in the complex projective plane.