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In this paper, we are interested in the motion estimation of a planar free-form object whose boundary is represented by a plane polynomial curve. We develop new Riccati equations in real variables for the slope and intercept parameters of the line factors obtained from the decomposition of the closed-bounded polynomial curve. We also derive new recursions on multiplicative scalars of the decomposition. For the estimation of motion parameters, an adaptive scheme has been employed and the results are verified by simulations.