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Implicit polynomial 2D curves and 3D surfaces are potentially among the most useful object or data representations for use in computer vision and image analysis. That is because of their interpolation property, Euclidean and affine invariants, and Bayesian recognizers. This paper studys and compares various fitting algorithms in a unified framework of stability analysis. It presents a new robust 3L fitting method that is repeatable, numerically stable and computationally fast and can be used for high degree implicit polynomials to capture complex object structure. With this, we lay down a foundation that enables a technology based on implicit polynomial curves and surfaces for applications to indexing into pictorial databases, robot vision, CAD for free-form shapes, etc.