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 − vt
z′ = z − ws
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.