Geometrically, motion takes place in a three-dimensional Euclidean space with a one-dimensional parameter. Let **σ** be a position vector in the space and *π* be a value of the parameter. Then **σ**(*π*) represents the positions of a particle in motion with the parameter *π* and the position **σ**.

There are two measures of the extent of motion: length and duration. A three-dimensional Euclidean space with a metric of length is called a *length space*. The parameter of motion for a length space is called *time*, which is the value of an independent duration. Let **x** be a position vector in length space and *t* be the time parameter. Then **x**(*t*) represents the positions of a particle in motion with the parameter *t* and the position **x**.

A three-dimensional Euclidean space with a metric of duration is called a *duration space*. The parameter of motion for a duration space is called *distance*, which is the value of an independent length. Let **z** be a position vector in duration space and *s* be the distance parameter. Then **z**(*s*) represents the positions of a particle in motion with the parameter *s* and the position **z**.

In the standard configuration, the Galilei (or Galilean) transformations of length and duration space are:

**x′** = **x** − **v***t*

**z′** = **z** − **w***s*

In many cases, length space with time is considered separately from duration space with distance. In other cases such as the wave equation they are considered together.