The existence of some automorphisms on eertain Laguerre plane L is used to deterraine the algebraic structure of the projective plane, which is a completion of the affine plane derived at a known point A. İn this paper, firstly a special Laguerre automorphisnı, that is a Laguerre inversion, is defined; and then the set of fixed points and the set of fixed circles of the Laguerre inversion are investigated.