# Maths - Stereographic 1D Derivation

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

 x y
= λ
 x' y'
 The line is (1): y' = 1 The circle is (2): x²+(y-½)²=(½)²
 from (2): x² = (1-(2y-1)²)/4 = (1-4y² +4y -1)/4 = -y² +y
from (1):
y =
 xy' x'
since y'=1
y =
 x x'
x' =
 x y
=
 √(y-y²) y
√((1/y)-1)

combining gives:

y=
 x x'
=
 1 x²+ 1      Roger Penrose - The Road to Reality: Partly a 'popular science' book as it tries to minimise the number of equations (Not that I'm complaining much his book 'Spinors and space-time' went over my head in the first few pages) it still has lots of interesting results that its difficult to find elsewhere.       Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (Cambridge Monographs on Mathematical Physics) by Roger Penrose and Wolfgang Rindler - This book is about the mathematics of special relativity, it very quickly goes over my head by I hope I will understand it one day.