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This video (at 44:00) says that in a projective space if three points are collinear, and two of those points lie at infinity, then the third point will also have to lie at infinity.

I wonder why that has to be true. Why can I not have three collinear points such that two of them are points of intersection of parallel lines (and hence lie at infinity), and the third is, say, the origin?

EDIT: One may think that the two points at infinity are the same point. However, what if we take one pair of parallel lines, and take points of intersection at both sides? I feel both these points are different, unless it is defined that both are the same.

user123
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Your edit explains your issue. In projective geometry, every line contains either exactly one point at infinity, or the line consists of ALL the points at infinity. Moreover, if two lines are parallel (in the classical sense), then when we think of them projectively, they meet at a point at infinity. From a very elementary standpoint, the point at infinity "remembers" the slope of the lines that go through it. But it does NOT remember which end of the line it's on. There's EXACTLY one point at infinity on each "normal" line.

(This is where the interesting topology of the projective plane comes from. If you take the plane and glue in one single point at infinity, you get a sphere. But the projective plane has lots of points glued in at infinity - a whole line's worth. So it's just different from the sphere.)

But at any rate, this means that if there are at least two points at infinity on a line, it can't be a "normal" line - it has to be the weird line at infinity.

coolpapa
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