Detecting similarity of rational plane curves

Abstract

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.

Detecting symmetries in polynomial Bézier curves

“…Therefore, the procedures in the Bézier form are not only simpler and more intuitive but also computationally cheaper than their O(n 3 ) counterparts [1] in the power basis. In 2D, another alternative would be to employ recent algorithms [30] for detecting similarity of rational curves, particularized to the polynomial case and self-similarity, but they are even more complex and expensive.…”

The problem of detecting when two implicit plane algebraic curves are similar

“…This explains why we assumed the equality of the degrees of C 1 , C 2 at the beginning of the paper. Furthermore, whenever C 1 , C 2 are not lines or circles the number of similarities between them is finite [1]; in fact, if C 1 , C 2 are not symmetric then there exists at most one similarity mapping one onto the other [1]. A similarity can either preserve or reverse the orientation.…”

Symmetries and similarities of planar algebraic curves using harmonic polynomials

“…equal up to a similarity transformation. Both problems have been addressed in many papers coming from applied fields like Computer Aided Geometric Design, Pattern Recognition or Computer Vision; the interested reader can check the bibliographies of the papers [1,2,5] for an exhaustive list. In fields like Patter Recognition or Computer Vision the problem of detecting similarity is essential because objects must be recognized regardless of their position and scale.…”

“…The papers [4,11] address symmetries of planar algebraic curves; in [4] the curve is assumed to be defined by a rational parametrization, while in [11] the input is defined by means of an implicit equation. Similarities of planar algebraic curves are considered in [5], where the input is a pair of rational parametrizations, [2], where the input is a pair of implicit equations, as in our case, and [9], where the input is also a pair of rational parametrizations. Nevertheless, the problem addressed in [9] is more general, since the authors consider how to deterministically recognize whether two rational curves of any dimension are related by some, non-necessarily orthogonal, projective or affine transformation.…”