Symmetries and similarities of planar algebraic curves using harmonic polynomials

TL;DR

This paper introduces efficient algorithms for detecting symmetries and similarities of planar algebraic curves, leveraging harmonic analysis and the properties of the Laplacian operator to simplify the problem.

Contribution

It presents novel deterministic algorithms that reduce the symmetry and similarity detection problems to harmonic cases, improving computational efficiency.

Findings

Algorithms are efficient and deterministic.

Symmetry detection reduces to harmonic case analysis.

Similarity checking leverages Laplacian commutation properties.

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 operator commutes with orthogonal transformations, and on efficient algorithms to find the symmetriessimilarities of a harmonic algebraic curvetwo given harmonic algebraic curves. In fact, we show that in general the problem can be reduced to the harmonic case, except for some special cases, easy to treat.