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 − *β*²):

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* (stanceline), lenticity *u*, and maximum lenticity 1/*c*, with *ζ* = *cu* and *λ* = 1/√(1 − *ζ*²):

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* (stance line), *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* → ∞: