In general, Lie (1871) showed that the conformal point transformations in R (composed of motions, similarities, and transformations by reciprocal radii) correspond in R to those sphere transformations which are contact transformations. Klein (1893) pointed out that by using minimal projection on hexaspherical coordinates, the 15-parameter Lie sphere transformations in R are simply the projections of the 15-parameter conformal point transformations in R, whereas the points in R can be seen as the stereographic projection of the points of a sphere in R.