<p><b>Abstract</b>—New representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a <it>complete set of invariants</it>, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a new centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in <tmath>$x,y$</tmath> measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the <tmath>$x,y$</tmath> data space. By recasting this approach in terms of a complex variable, i.e., <tmath>$x+iy=z$</tmath>, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.</p>