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 homogenous 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 study a certain complex representation of the polynomial, 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 our representation allows the detection of rotation symmetry simply by looking at the indices of all non-vanishing complex parameters.