It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often considered a special trick for analyzing four-bar motion, are in fact uniquely suited to two new techniques for analyzing polynomial systems: the BKK bound and the product-decomposition bound. From this synergistic viewpoint, a fundamental formulation of planar kinematics is developed and used to prove several new results, mostly concerning the degree and circularity of the motion of planar linkages. Useful for both analysis and synthesis of mechanisms, the approach both simplifies theoretical proofs and facilitates the numerical solution of mechanism problems.

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