# Affine and projective geometry properties and differences

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.

• – asdf Jan 4 '18 at 11:07
• 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 – 57Jimmy Jan 4 '18 at 11:08
• 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). – Morgan Rodgers Jan 5 '18 at 5:58

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.