Some new formulae for the derivative bounds of parametric curves, such as the general polynomial curves and rational polynomial curves in CAGD, are presented. Based on these new formulae, the point-by-point algorithm for rasterizing parametric curves is developed in this paper. To solve the problem of repetition and discontinuity arising from the previous algorithms, a new rule of interpolation is given. Without doubt, these results will remarkably improve the efficiency of modeling, intersection, approximation, rendering and rasterizing of curves.