Dual Euclidean transformations are required to transform six dimensions of length and duration: one Euclidean transformation for length space with time and one Euclidean transformation for duration space with distance. The two Euclidean transformations are:

**x′** = **x** − **v***t* and **z′** = **z** − **w***s*

where **x** and **x′** are length space vectors, *t* is the time parameter, **z** and **z′** are duration space vectors, and *s* is the distance parameter. The two transformations are related by the rod-clock rate:

**x** = *c***z** and *s* = *ct*

where *c* is the rod-clock speed. The two Euclidean transformations are related by the rod-clock speed:

**x′** = **x** − **v***t* = *c***z′** = *c***z** − **v***s*/*c* = *c***z** − **β***s*

where **β** = **v**/*c*. We then have

**z′** = **z** − **v***s*/*c*² = **z** − **w***s*

where **w** = **v**/*c*². That is, *c***w** = **v**/*c* = **β**.

Note there is some similarity with the Lorentz transformations without the gamma factor.