Page "Lenstra elliptic curve factorization" Paragraph 25

Before we consider the projective plane over /~, first have a look at the ' normal ' projective space over.

Now, instead of studying the points, the lines through the origin will be studied.

The line can be represented a non-zero point if you use the equivalence relation ~ on it, given by: if and only if there exists a non-zero number such that, and.

Due to this equivalence relation the space will be called a plane.

In the projective plane, points, denoted by, are ' the same ' as lines in a threedimensional space that go through the origin.

Remark that the point does not exist here, because these does not represent a line.

Now we observe that almost all lines go to the-plane, except from the lines parallel to this plane.

This lines are in the projective plane the ' points of infinity ' that are used in the affine-plane above.