Isomorphisms
A gyrovector space isomorphism preserves gyrogroup addition and scalar multiplication and the inner product.
The three gyrovector spaces Möbius, Einstein and Proper Velocity are isomorphic.
If M, E and U are Möbius, Einstein and Proper Velocity gyrovector spaces respectively with elements vm, ve and vu then the isomorphisms are given by:
| EU by |
| UE by |
| EM by |
| ME by |
| MU by |
| UM by |
From this table the relation between and is given by the equations:
This is related to the connection between Möbius transformations and Lorentz transformations.
Read more about this topic: Gyrovector Space