This page looks at a one dimensional case of stereographic projection for a more general discussion of stereographic projection see page here.
Here we look at a one dimensional euclidean space embedded in a two dimensional projective space, we are using the stereographic model to do this projection. 

In one dimensional projective space using stereographic model:
 A straight lines is mapped to a circle in projective space.
 The point at infinity is given by θ=180 degrees.
Derivation for translation between projective (stereographic) and euclidean spaces
The line is (1): 
y' = 1 
The circle is (2): 
x²+(y½)²=(½)² 
from (2): 
x² = (1(2y1)²)/4
= (14y² +4y 1)/4
= y² +y 
combining gives: