In this paper, we present a new curve-fitting algorithm for vectorizing hand-drawn key frames in a computer-aided cartooning system. In our algorithm, a new optimization technique is developed for finding a parametric curve, which approximates to the given data points. In order to obtain the desired results, we also define a new distance function to measure the error between the curve and the data points. In addition, in the piecewise curve fitting our method can satisfy the geometrical continuity on a non-corner knot. With these techniques the hand-drawn images can faithfully be represented by piecewise cubic Bézier curves.