Confusion about the point at infinity with respect to inversion in geometry.

Question

I was studying inversion in Olympiad geometry, and they (Evan Chen's book EGMO) mentioned that we can extend the Euclidean plane by adding a point $P_{\infty}$ such that each line passes through it, and no circle passes through it.

The reason for this they said was that now two parallel lines meet at that point only, and the center can go there on inversion.

But now I have a really stupid confusion:

Does this mean that all non parallel lines meet at two points and parallel lines meet at only one?

I am very confused by this part now, I tried looking up some things on Wikipedia but they had defined very different things and it just made me more confused.

I would really appreciate if anyone could clear this really dumb doubt of mine,

Thank you!

Answer 1

You asked

Does this mean that all non parallel lines meet at two points and parallel lines meet at only one?

I think that the Wikipedia article Stereographic projection will make this visually clear. In this projection, all lines and circles in the plane of projection come from circles on the surface of the sphere. All of the circles that pass through the pole project down onto straight lines in the plane while all of the circles that do not pass through the pole project down onto circles in the plane. Any two distinct spherical circles are either disjoint, or are tangent at one point, or else have two points of intersection just as circles do in the plane.

Thus, any two distinct circles that pass through the pole already intersect at the pole. If they are tangent then that is the only intersection point and they project onto two parallel lines, otherwise they intersect at another point and they project down onto two intersecting lines.

Note that the pole on the sphere corresponds to the ideal point $P_\infty$ which is the only point on the sphere which does not project down to a point in the plane.

Answer 2

Does this mean that all non parallel lines meet at two points and parallel lines meet at only one?

Yes, for any two distinct lines, that would be the case. You're trading the axiom that "two distinct points determine a line" for a different one where "three distinct points determine a 'cline'" (I thought they called them "lircles" actually...)

You are not doing Euclidean geometry anymore, but have passed to Möbius geometry. I suppose the chapter you are reading teaches you how to transition between the two.