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We propose a new, simple theory-based, accurate polarization microscope for birefringence imaging of cytoskeletal structures of biological cells. The new theory lets us calculate very easily the phase retardation and the orientation of the principal axis of a particular area of a biological living cell in media by simply measuring the intensity variation of a pixel of a CCD camera while rotating a single polarizer. Just from the measured intensity maxima and minima, the amount of phase retardation δ between the fast and the slow axis of the sample area is obtained with an accuracy of 5.010±0.798×10-3 rad. The orientation of the principal axis is calculated from the angle of the polarizer for the intensity maximum. We have compared our microscopes with two previously reported polarization microscopes for birefringence imaging of cytoskeletal structures and demonstrated the utility of our microscope with the phase retardation and orientation images of weakly invasive MCF7 and highly invasive MDA MB 231 human breast cancer cells as an example.