This paper proposes a new geometric Newton-Raphson method for dealing with a rational polynomial curve. The algorithm is robust and at the same time locally unique. Although rational polynomial curves and surfaces have become standard forms in computer-aided design, they have many problems. For example, a Newton-Raphson algorithm for dealing with a rational polynomial curve tends to be unstable. This is a fatal problem. We propose to homogenize the coordinates of a rational curve when it is applied to the Newton-Raphson algorithm. Then it becomes very robust. Furthermore the solution point becomes locally unique with respect to an initial parameter range when the parameter is also homogenized in addition to the coordinates, because with this technique we have a freedom of controlling parameter values and we can adjust the increment of the parameter appropriately.