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. stereographic projection

In one dimensional projective space using stereographic model:

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

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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag 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.

 

cover 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.

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