Geometry
Fundamental to the idea that ∞ is a point no different from any other is the way the real projective line is a homogeneous space, in fact homeomorphic to a circle. For example the general linear group of 2×2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations, with the understanding that when the denominator of the linear fractional transformation is 0, the image is ∞.
The detailed analysis of the action shows that for any three distinct points P, Q and R, there is a linear fractional transformation taking P to 0, Q to 1, and R to ∞. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant.
The terminology projective line is appropriate, because the points are in 1-1 correspondence with one-dimensional linear subspaces of R2.
Read more about this topic: Real Projective Line
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