The projection operators that project the potential energy of electron interactions onto the lowest Landau level of a fractional quantum Hall state are considered for both disk and spherical geometries. The study has been performed by using the Laughlin, the quasi-hole and the quasi-electron wave functions. In the case of spherical geometry, the form of the coordinate-operator transformation obtained in a recent article has been amended. The results obtained are expected to be useful in the calculation of the quasi-hole energy, the quasi-electron energy and the energy gap.