Thread: Surjection from set of all lines to real projective space

can anyone think of a natural surjection from the set of all lines in R^2 to the Real projective space RP1 ????

Ιt is mentioned in every definition!

There are several equivalent definitions of RP1. What definition are you using?

Well...
the geometric one?
Wiki says RP^1=(R^2-{0})/~, where ~ is the "same line through the origin" equivalence relation.


The question goes "find a natural surjection",
so the keyword "natural"
demands that every line in R^2 be identified as an element of RP^1.

Ofcourse, if that line is through the origin, the identification is obvious.
If it is not, the slope provides an equivalence relation of all lines in the plane, and there is a unique representative of each class to pass through the origin.