Plotting algebraic curves in the projective plane

As a follow-up to the question Are there any good computer programs for drawing (algebraic) curves?, are there any programs that can plot real algebraic curves in (a model of) the projective plane?

flag	asked Oct 20 2010 at 20:57 lhf 1,875●1●8●19

I'm not sure what the point of this would be. You can tell what the points at infinity on the curve are by looking at the slopes of the asymptotes. Alternately, you can just change affine charts to look at the original curve from a different perspective. – Jack Huizenga Oct 20 2010 at 21:27

@Jack, sure you use different charts. But I'd like to see $y=x^2$ as an ellipse in a single drawing, for instance. – lhf Oct 20 2010 at 21:31

If you model the projective plane as $S^2/{+-1}$ then curves are represented by their preimages on the sphere. The program "surfex" which I advertised in the earlier question has a build in function to plot intersections of hypersurfaces (e.g. a sphere and a cone). – Heinrich Hartmann Oct 20 2010 at 21:33

@Heinrich, thanks. Please add your comment as an answer. – lhf Oct 20 2010 at 22:04

If you make the change of coordinates $z \mapsto z+y$ and $y \mapsto z-y$, the equation transforms in to $x^2 + y^2 = 1$, a circle, which you can draw in a plane viewed as a piece of the projective plane. This example seems reasonable as we get an `ellipse' as we wanted. However, I think the fact that you are looking at the real locus only makes this rather artificial... – David Holmes Oct 21 2010 at 9:35

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