What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.
For the (3+1) Lorentz transformation there are length spaceaxes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):
The symmetric form is
The symmetric Lorentz transformation for vectors is
For the (1+3) dual Lorentz transformation there are duration space axes x, y, and, z; stantial axis r (placeline), lenticity u, and maximum lenticity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):
The dual symmetric form is
The dual symmetric Lorentz transformation for vectors is
For the (3+1) Ignatowski transformation there are length space axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity V, with β = v/V:
The symmetric form is
For the (1+3) dual Ignatowski transformation there are duration space axes x, y, and, z; stantial axis r (placeline), lenticity u, and maximum lenticity U, with ζ = u/U:
The dual symmetric form is
The Galilean transformation has V → ∞:
The dual Galilean transformation has U → ∞:
