Euclidean stratum

For the sake of completeness, Euclidean geometry is briefly discussed. It does not differ much from metric geometry as we have defined it here. The difference is that the scale is fixed and that therefore not only relative lengths, but absolute lengths can be measured. Euclidean transformations have 6 degrees of freedom, 3 for orientation and 3 for translation. A Euclidean transformation has the following form

$${\bf T}_E \sim \left[\begin{array}{cccc} r_{11} & r_{12} & r... ... & r_{32} & r_{33} & t_Z \\ 0 & 0 & 0 & 1 \end{array}\right]$$ (B39)

with $r_{ij}$ representing the coefficients of an orthonormal matrix, as described previously. If ${\bf R}$ is a rotation matrix (i.e. det${\bf R}=1$) then, this transformation represents a rigid motion in space.