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I've being studying projective geometry for a few days. One of the key aspect of this geometry is that any pair of lines always intersects at some unique point. For instance, the lines $2x + y = 0$ and $4x + 2y + 1 = 0$ of $\mathbb{RP}^2$ intersects at a unique point with homogeneous coordinates $[2:1:0]\times[4:2:1]=[1:-2:0]$. My question is the following : do we also have this property in affine geometry, i.e. does any par of lines always intersects at some unique point in affine geometry?

I can guess the answer would be no but I would not be able to justify it correctly. After working with these two geometries for a few days I still fail to see the differences between them. Any help would be much appreciated.

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    $\begingroup$ See this math.stackexchange.com/questions/264857/… $\endgroup$ – asdf Jan 4 '18 at 11:07
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    $\begingroup$ No, that's one of the reasons for introducing projective geometry. Affine geometry is more or less the usual Euclidean geometry, so parallel lines do not intersect $\endgroup$ – 57Jimmy Jan 4 '18 at 11:08
  • $\begingroup$ Your property of lines always intersecting in a unique point is a property of projective planes, and does not hold for higher dimensional projective spaces (you can have non-intersecting lines of $\mathbb{RP}^{3}$, for example). $\endgroup$ – Morgan Rodgers Jan 5 '18 at 5:58
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does any par of lines always intersects at some unique point in affine geometry?

No, if the lines are parallel, then in affine geometry they do not intersect at all. In projective geometry, there are points “at infinity” for every bundle of parallel lines.

Difference between Projective Geometry and Affine Geometry might be of interest to you.

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Consider the two lines of your example. An intersection point in the affine plane $\mathbb{R}^2$ is just a couple of real numbers $(x,y)$ that satisfy both equations. But if $2x+y=0$, then $4x+2y+1 = 1 \neq 0$, so there can be no intersection point.

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