Symmetries and similarities of planar algebraic curves using harmonic polynomials

Abstract

We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether 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 commutes with orthogonal transformations, and on efficient algorithms to find the symmetries ∕ similarities of a harmonic algebraic curve ∕ two given harmonic algebraic curves. In fact, we show that, except for some special cases, the problem can be reduced to the harmonic case.