# affine vs. projective conics on their specified fields

I am needing to make a detailed comparison of the affine conics in $\mathbb{R}^2$ with that of the projective conics in $\mathbf{P}^2$

could only identify:

a) classification of non-degenerate conics

b) single pt given by $x^2 + y^2 = 0$

c) line $xy=0$

(e,f,g) empty set given by $x^2 + y^2 = -1$, $x^2 = -1$ or $0 = 1$

j) parallel lines $x(x-1)=0$

k) double line $x^2 = 0$

l) plane given by $0=0$

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Projective over what field? –  Qiaochu Yuan Aug 23 '12 at 6:01
Okay. This looks like homework; what have you tried? –  Qiaochu Yuan Aug 23 '12 at 6:14
–  robjohn Sep 9 '12 at 1:08
@ZhenLin: I've removed the close flags on this question since its duplicate has been deleted. –  robjohn Sep 9 '12 at 1:20

Projectively, all nondegenerate conics over the reals are equivalent. In the affine plane, you can determine whether a conic is an ellipse, a parabola or a conic by counting the points at infinity: hyperbolas have two of them, the parabola one, and ellipses none (no real ones in any case).

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I edited my problem better so please check it out above. Thanks –  mary Sep 5 '12 at 7:13
Franz: I would count $x^2 + y^2 + z^2 = 0$ as a nondegenerate real projective conic. Since it has no real points, it is certainly not projectively equivalent to $x^2 + y^2 - z^2 = 0$. In this standard arithmetic-geometric philosophy, there are precisely two equivalence classes of nondegenerate projective conics over $\mathbb{R}$. –  Pete L. Clark Sep 9 '12 at 1:28