The NURBS curves and surfaces have become the standard descriptions in the field of CAD and computer graphics. They have several problems, however, such as numerical instability, limited convex hull property, and inefficiency of computing points on a curve or a surface. In this paper, it is shown that these problems can be resolved by using our newly proposed homogenized NURBS. In the homogenized NURBS, the sum of the blending functions need not be unity and the fractional representation of the functions is replaced by the integral representation. By taking advantage of this characteristic and homogeneous curves, an extended finite difference method which efficiently and accurately computes points on a curve is presented. By using the homogenized NURBS, numerical instability vanishes, convex hull property holds strictly regardless of the signs of weights, and exact points on the curve can be computed very efficiently.