In R the Laguerre group leaves invariant the relation d x 2 + d y 2 − − d r 2 , which can be extended to arbitrary R as well. For instance, in R it leaves invariant the relation d x 2 + d y 2 + d z 2 − − d r 2 . This is equivalent to relation d x 2 + d y 2 + d z 2 + d r 2 in R by using minimal (isotropy) projection with imaginary radius coordinate, or cyclographic projection (in descriptive geometry) with real radius coordinate. The transformations forming the Laguerre group can be further differentiated into "direct Laguerre transformations" which are related to motions preserving both the tangential distance as well as the sign; or "indirect Laguerre transformations" which are related to orientation-reversing motions, preserving the tangential distance with the sign reversed. The Laguerre inversion first given by Laguerre in 1882 is involutory, thus it belongs to the indirect Laguerre transformations. Laguerre himself did not discuss the group related to his inversion, but it turned out that every Laguerre transformation can be generated by at most four Laguerre inversions and every direct Laguerre transformation is the product of two involutory transformations, thus Laguerre inversions are of special importance because they are generating operators of the entire Laguerre group.