When used in the on-the-grid solvers of the stationary or time-dependent Schrodinger equation, coordinate mapping allows one to achieve a very accurate description of the wave function with an optimal number of the grid points. The efficiency of the mapped Fourier grid methods has been recently demonstrated by V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws [J. Chem. Phys. 110, 9865 (1999)] and by D. Lemoine [Chem. Phys. Lett. 320, 492 (2000)]. In this paper we propose a discrete coordinate representation based on a numerical mapping in cylindrical and spherical coordinates. Within proposed approach, the Hamiltonian matrix is Hermitian, and the use of the fast cosine and sine Fourier transforms provides a very efficient way of calculating the Laplacian operator.