A Laguerre transformation (L-transform) is a mapping which is bijective on the sets of oriented planes and oriented spheres, respectively, and preserves tangency between plane and sphere. L-transforms are more easily understood if we use the so-called cyclographic model of Laguerre geometry. There, an oriented sphere S is represented as point S := ⁡ ⁡ ( m , R ) ∈ ∈ R 4 . An oriented plane P in E 3 may be interpreted as the set of all oriented spheres which are tangent to P . Mapping P via this set of spheres into R 4 , one finds a hyperplane in R 4 which is parallel to a tangent hyperplane of the cone x 1 2 + x 2 2 + x 3 2 − − x 4 2 = 0 . In the cyclographic model, an L-transform is seen as a special affine map (Lorentz transformation),...