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In this paper, we introduce a completely new approach to fitting implicit polynomials to data, and to studying these polynomials. The power of these models is in their ability to represent non-star complex shapes in two- (2D) and three-dimensional (3D) data, to permit position-invariant shape recognition based on new complete sets of Euclidean and affine invariants, and to permit fast, stable single-computation pose estimation. The algorithm represents a significant advancement of implicit polynomial technology for four important reasons. First, it is orders of magnitude faster than existing fitting methods, and the algorithms for 2D and 3D are essentially the same. Second, it has significantly better repeatability, numerical stability and robustness than current methods in dealing with noisy, deformed, or missing data. Third, it can easily fit polynomials of high, such as 14th or 16th, degree. Fourth, additional linear constraints can be easily incorporated into the fitting process, and general linear vector space concepts apply.