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 operator 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 in general the problem can be reduced to the harmonic case, except for some special cases, easy to treat.

Approximate symmetries of planar algebraic curves with inexact input

“…The elements of a finite symmetry group are rotations (all of them with the same center) and reflections (axes of all of them passing through the same point). Let us recall the following statement, which can be efficiently used to verify whether φ ∈ Sym(C), see [4] for more details: Then analogously to Sym(C) we can write that φ ∈ Sym(f ), as well. When there is no danger of confusion we will not distinguish between the symmetries of C and f .…”

“…Once the approximate symmetric curve is known then we can follow any exact approach for computing symmetries of curves; e.g. a method from [4] which has been formulated recently. This exact method works with the observation that it is not easy, in general, to find symmetries φ belonging to Sym(C) directly and one has to apply a suitable alternative approach -for instance to find some new polynomial h(x, y) such that Sym(h) is finite, easy to determine (i.e., easier then Sym(f )) and Sym(C) = Sym(f ) ⊂ Sym(h).…”

“…This exact method works with the observation that it is not easy, in general, to find symmetries φ belonging to Sym(C) directly and one has to apply a suitable alternative approach -for instance to find some new polynomial h(x, y) such that Sym(h) is finite, easy to determine (i.e., easier then Sym(f )) and Sym(C) = Sym(f ) ⊂ Sym(h). In [4], a successive application of the Laplace operator yielding the sequence…”

“…Then the goal is to decide whether two given geometric objects are related by some (projective, affine, similar, or isometric) transformation and in the affirmative case to detect all such equivalences. These problems are often addressed in papers coming from applied fields like Computer Aided Geometric Design, Pattern Recognition or Computer Vision, see [2][3][4] for the exhaustive list of references. In fields like Patter Recognition or Computer Vision especially the problem of detecting similarity is essential because objects must be recognized regardless of their position and scale.…”

“…not two implicitly given, planar algebraic curves are similar, i.e., equal up to a similarity transformation, was considered in [4]. A problem of computing projective equivalences of special algebraic varieties, including projective (and other) equivalences of rational curves was investigated in [12].…”