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For rigid-body guidance mechanism synthesis through four finitely separated positions and four mixed positions, a displacement matrix method was presented for calculating the Burmester curves. After the equations of Burmester curves have been deduced, they were turned into the standard forms through coordinate transformations. Then according to the projective geometry theory that two pencils-of-low-order curves constituted a high-order curve, the Burmester curves could be single-valuably and orderly expressed by intersections of a pencil-of-circles and a pencil-of-lines. Finally, solution regions were set up so that the designers could have an intuitional and exact knowledge of mechanism properties.