Wikipedia:Reference desk/Archives/Mathematics/2014 January 5

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January 5

Conic Sections expressed in ${\widehat {\mathbb {R} }}$ x ${\widehat {\mathbb {R} }}$

Since in regards to the concept of conic sections hyberbolas can sort of be viewed as ellipses that have gone through infinity and come out the other side, does it become easier to define the conic sections in ${\widehat {\mathbb {R} }}$ x ${\widehat {\mathbb {R} }}$ since in that regard it sort of makes sense to view a parabola with one focus at Infinity and on beyond infinity for the Hyberbolas? — Preceding unsigned comment added by Naraht (talk • contribs) 05:51 5 January 2014 (UTC)

No. The appropriate context for this is the projective plane, in which the "points at infinity" form a line, the "line at infinity." Given one of these points P at infinity, the set of lines passing through it is a family of parallel lines. So you can think of there being one point at infinity for each possible direction a line in the plane can have; all lines in that direction meet at that point. If you think about the example of a hyperbola, you then see why the points at either end of the same asymptote are actually the same. But there are two points at infinity on the hyperbola, corresponding to the intersections of the hyperbola with each of its two asymptotes. This means the line at infinity intersects the hyperbola in two different points, which is what often happens when you take the intersection of a line and a conic section. Ellipses do meet some lines, but not the line at infinity. On the other hand, a parabola is tangent to the line at infinity. In homogeneous coordinates, the equation of a conic section is ax² + by² + cz² + dxy + exz + fyz = 0 96.46.198.120 (talk) 11:28, 5 January 2014 (UTC)

I agree that the projective plane is a good context, and the description here is correct. However, as one moves one focus in the projective plane through the line at infinity so that the conic intersects with the line at infinity, the focus reappears from the opposite direction with what was the arc of the ellipse "flipped over". So the description as "can sort of be viewed as ellipses that have gone through infinity and come out the other side" is valid, keeping in mind that in the projective context, "beyond infinity" is a finite distance away in the opposite direction on the same line. As to whether it makes it easier, I can't comment. —Quondum 19:46, 5 January 2014 (UTC)