A general ray tracing scheme has been developed for using a personal computer which trace finite rays through any non-rotationally symmetric system. This scheme may be used for the surface type such as conic section with or without aspherics, toric surfaces, sagittal and tangential cylindrical sections and axicons. Specially, any combinational of decentered, tilted and rotated surfaces has been considered. Before transfering to the next surfaces, the local coordinates are refered back to an initial reference coordinate system. We can get a mathmtical model of a non-rotationally symmetrical finite ray trace running on an inexpensive personal computer.