We recently developed and reported an eye fixation monitor that detects the fovea by its radial orientation of birefringent nerve fibers. The instrument used a four-quadrant photodetector and a normalized difference function to check for a best match between the detector quadrants and the arms of the bow-tie pattern of polarization states surrounding the fovea. This function had a maximum during central fixation but could not tell where the subject was looking relative to the center. We propose a linear transformation to obtain horizontal and vertical eye position coordinates from the four photodetector signals, followed by correction based on a priori calibration information. The method was verified on both a computer model and on human eyes. The major advantage of this new eye-tracking method is that it uses true information coming from the fovea, rather than reflections from other structures, to identify the direction of foveal gaze.