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I understand parallel lines in Euclidean space intersect at the line at infinity in terms of projective space.

My question is for a single line. A single line if extended to infinity must intersect the line at infinity at some point (correct me if this wrong.). The thing that I find hard to interpret is how could it not intersect the infinity line in two points which are located in the two opposite directions of the line?

I have checked this existing question line at infinity and it didn't help.

Community
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Stack crashed
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2 Answers2

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Each line has just one point at infinity, which is approached by going in either direction along the line. Two lines share the same point at infinity if and only if they are parallel to each other. Two lines not parallel to each other have different points at infinity.

When one adds to the affine line a point at infinity that is approached by going in either direction, the line becomes topologically a circle.

Michael Hardy
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  • Is this a valid interpretation - a topological circle would allow to revisit the same point if one continues along this line for infinite amount of time (non zero speed)? – Stack crashed Apr 16 '16 at 16:31
  • $\ldots,{}$or even a finite amount of time: for example, let $\theta$ move along the real line $\mathbb R$, and observe what $\tan\theta$ does as $\theta$ goes from $-\pi/2$ to $\pi/2$. Its value is $\infty$ at those two points. $\qquad$ – Michael Hardy Apr 16 '16 at 16:33
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Just speaking in "intuitive" terms,
consider three lines intersecting at one point. Keep one line firm, tilt one of the line, making the intersection to move away towards infinity, in one of the directions, slide the third line parallel to itself to keep the one-point intersection. Continuing to tilt beyond the parallelism of the first two lines, the one point intersection and the third line reappear from the other direction. This might help to fill the intuition gap.

G Cab
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