Complex representations of algebraic curves

We employ a complex representation for an algebraic curve, and illustrate how the algebraic transformation which relates two Euclidean equivalent curves can be determined using this representation. The idea is based on a complex representation of 2D points expressed in terms of the orthogonal x and y variables, with rotations of the complex numbers described using Euler's identity. We develop a simple formula for integer multiples of the rotation angle of the Euclidean transformation in terms of the real coefficients of implicit polynomial equations that are used to model various 2D free-form objects. When there is a translation, it can be determined in a straightforward manner using an estimation of the rotation angle and some new results on conic-line decompositions.