Maths - Hemisphere 2D Derivation

Here we look at a two dimensional euclidean space embedded in a three dimensional projective space, we are using the hemisphere model to do this projection.

hemisphere projection

In two dimensional projective space using hemisphere model:

From the above we can see that, as we move away from the origin, lines that cross at 90° in euclidean space do not cross at 90° in projective space. So angles away from the origin are not conserved by the projection.

horizontal lines

vertical lines

Derivation for translation between projective (hemisphere) and euclidean spaces

Since we are projecting using a straight line then there is a linear relationship between a point on the plane and its projection on the sphere:

x
y
z
= λ
u
v
1

where:

Dividing the top two rows by the bottom row gives the euclidean coodinates in terms of the projective coordinates:

u
v
=
x/z
y/z

To go the other way from the projective coordinates to the euclidean coodinates we need to use the information that the coordinates are constrained to a unit sphere so:

x² + y² + z² = 1

rearanging gives:

x²/z² + y²/z² + 1 = 1/z²

u² + v² + 1 = 1/λ²

λ = ±1/√(u² + v² + 1)

substituting this into the first vector equation gives:

x
y
z
= ±1/√(u² + v² + 1)
u
v
1

 


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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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