Real projective line
It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity".
The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line.
Each element of PGL(2, R) can be defined by a nonsingular 2×2 real matrix, and two matrices define the same element of PGL(2, R) if one is the product of the other and a nonzero real number.
Topologically, real projective lines are homeomorphic to circles.
The starting point is a real vector space of dimension 2, V. Define on V ∖ 0 the binary relation v ~ w to hold when there exists a nonzero real number t such that v = tw.
The definition of a vector space implies almost immediately that this is an equivalence relation.
The real projective line P(V) is the set of all equivalence classes.
If one chooses a basis of V, this amounts (by identifying a vector with its coordinate vector) to identify V with the direct product R × R = R2, and the equivalence relation becomes (x, y) ~ (w, z) if there exists a nonzero real number t such that (x, y) = (tw, tz).
However, the fact that equivalence classes are not finite induces some difficulties for defining the differential structure.
Therefore, the projective line may be considered as the quotient space of the circle by the equivalence relation such that v ~ w if and only if either v = w or v = −w.
This embedding allows us to identify the point [x: y] either with the real number x/y if y ≠ 0, or with ∞ in the other case.
for homogeneous coordinates denotes the equivalence class of the row vector.
The elements of GL2(R) that act trivially on P1(R) are the nonzero scalar multiples of the identity matrix; these form a subgroup denoted R×.
Using the identification R ∪ ∞ → P1(R) sending x to [x:1] and ∞ to [1:0], one obtains a corresponding action of PGL2(R) on R ∪ ∞ , which is by linear fractional transformations: explicitly, since the class of
[4] Some authors use left action on column vectors which entails switching b and c in the matrix operator.
