  In this paper, we investigate the dynamics of both free particle and
isotropic harmonic oscillator constrained to move on a spheroidal surface using
two consecutive projections: a projection onto a sphere surface followed by the
gnomonic projection onto a tangent plane to the spheroid. We obtain the
Hamiltonian of the aforementioned systems in terms of the Cartesian coordinates
of the tangent plane and then quantize it in the standard way. It is shown that
the effect of non-sphericity of the surface can be treated as the appearance of
an effective potential. By using the perturbation theory up to the first order
in second eccentricity of the spheroid, we approximately calculate the
eigenfunctions and eigenvalues of the free particle, as well as the isotropic
harmonic oscillator on the spheroidal surface. We find that the deviation from
the sphericity plays an important role in splitting the energy levels of the
isotropic oscillator on a sphere, and lifting the degeneracy.
