Implicitization of parametric curves by matrix annihilation

Both parametric and implicit representations can be used to model 2D curves and 3D surfaces. Each has certain advantages compared to the other. Implicit polynomial (IP) methods are not as popular as parametric procedures because the lack of general procedures for obtaining IP models of higher degree has prevented their general use in many practical applications. In most cases today, parametric equations are used to model curves and surfaces. One such parametric representation, elliptic Fourier Descriptors (EFD) has been widely used to represent 2D and 3D curves, as well as 3D surfaces. Although EFDs can represent nearly all curves, it is often convenient to have an implicit algebraic description F(x,y)=0, especially for determining whether given points lie on the curve. Algebraic curves and surfaces also have proven very useful in many model-based applications. Various algebraic and geometric invariants obtained from these implicit models have been studied rather extensively. In this paper, we present a new non-symbolic implicitization technique called the matrix annihilation method, for converting parametric Fourier representations to implicit polynomial form.