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We present novel, deterministic, efficient algorithms to compute thesymmetries of a planar algebraic curve, implicitly defined, and to checkwhether or not two given implicit planar algebraic curves are similar, i.e.equal up to a similarity transformation. Both algorithms are based on the fact,well-known in Harmonic Analysis, that the Laplacian operator commutes withorthogonal transformations, and on efficient algorithms to find thesymmetriessimilarities of a harmonic algebraic curvetwo given harmonicalgebraic curves. In fact, we show that in general the problem can be reducedto the harmonic case, except for some special cases, easy to treat.

Symmetries and similarities of planar algebraic curves using harmonic polynomials