The literature deals with the modulation transfer function (MTF) only for object brightness distribution functions (OBDFs) oriented along the meridional and sagittal directions. This paper addresses computation of the geometrical MTF for an off-axis source point when the OBDF is oriented along any arbitrarily defined direction. This study finds that the MTF is not a monotonic increasing or decreasing function when the direction of the OBDF is changing. Consequently, the extreme MTF values may occur when the OBDF is aligned at any direction between the meridional and sagittal directions. Four theorems are provided for the MTF and the phase shift variations that take place when the OBDF is translated or rotated. It is found that the MTF and the phase shift are symmetrical or antisymmetrical about certain directions. Thus, to observe all possible changes in the MTF and the phase shift, it is sufficient to rotate the OBDF through a range of just $90°$ . The presented method is based on a recent irradiance method for MTF computation that does not rely on counting the number of ray hits on a mesh, making the method immune to effects of grid size and thus improving traditional accuracy.