As shown above, the group of conformal point transformations in R (composed of motions, similarities, and inversions) can be related by minimal projection to the group of contact transformations in R transforming circles or spheres into other circles or spheres. In addition, Lie (1871, 1896) pointed out that in R there is a 7-parameter subgroup of point transformations composed of motions and similarities, which by using minimal projection corresponds to a 7-parameter subgroup of contact transformations in R transforming circles into circles. These relations were further studied by Smith (1900), Blaschke (1910), Coolidge (1916) and others, who pointed out the connection to Laguerre's geometry of reciprocal directions related to oriented lines, circles, planes and spheres. Therefore, Smith (1900) called it the "group of the geometry of reciprocal directions", and Blaschke (1910) used the expression "Laguerre group". The "extended Laguerre group" consists of motions and similarities, having 7 parameters in R transforming oriented lines and circles, or 11 parameters in R transforming oriented planes and spheres. If similarities are excluded, it becomes the "restricted Laguerre group" having 6 parameters in R and 10 parameters in R, consisting of orientation-preserving or orientation-reversing motions, and preserving the tangential distance between oriented circles or spheres. Subsequently, it became common that the term Laguerre group only refers to the restricted Laguerre group. It was also noted that the Laguerre group is part of a wider group conserving tangential distances, called the "equilong group" by Scheffers (1905).