Implicitizing rational surfaces without base points by moving planes and moving quadrics

It was proven by Cox, Goldman and Zhang that a tensor product rational surface without base points can be implicitized by moving quadrics whenever the rational surface doesn't contain low degree moving planes following it. However, when the rational surface does have low degree moving planes, Cox, Goldman and Zhang's method fails. In this paper, we show that a rational surface without base points can always be implicitized by moving quadrics together with moving planes whether the rational surface has low degree moving planes or not. A specific method is also provided to construct the moving planes and moving quadrics that comprise a compact determinantal representation of the implicit equation of the rational surface.