Rotations, translations and symmetry detection for complexified curves

A plane algebraic curve can be represented as the zero-set of a polynomial in two—or if one takes homogeneous coordinates: three—variables. The coefficients of the polynomial determine the curve uniquely. Thus features of the curve, like for instance rotation symmetry, must find their correspondence in the algebraic structure of the coefficients of the polynomial. In this article we will investigate how one can extract geometric curve features from the algebraic description of the curve. In particular, we will use a certain complex representation of polynomials introduced by (Tarel, J.-P., Cooper, D.B., 1998. A new complex basis for implicit polynomial curves and its simple exploitation for pose estimation and invariant recognition. In: Conference on Computer Vision and Pattern Recognition (CVPR'98), pp. 111–117; Unel, M., Wolovich, W.A., 1998. Complex representations of algebraic curves. In: International Conference on Image Processing ICIP 1998, pp. 272–276), which is very appropriate for the task of feature detection. In this complex representation actions on the curve parameters induced by geometric rotations or translations of the plane become very simple. Invariant expressions in the complexified parameters and also normal forms are easily accessible. Furthermore this representation allows the detection of rotation symmetry simply by looking at the indices of all non-vanishing complex parameters.