 Synthetic geometry of the real projective plane

two parallel lines on the plane meet at a point on the sphere!

Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.

One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"

lines in the real projective plane are planes in ${\mathbb{R}}^3$.

For every two projective points there is a single projective line that passes through them.

Since it is a plane in ${\mathbb{R}}^3$, it always intersects the real plane at a line.

Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.