The fuzzy curve-tracing (FCT) algorithm can be used to extract a smooth curve from unordered noisy data. However, the model produces good results only if the curve shape is either opened or closed. In this paper, we propose several techniques to generalize the FCT algorithm for tracing complicated curves. We develop a modified clustering algorithm that can produce cluster centers less dependent on the pre-specified number of clusters, which makes the reordering of cluster centers easier. We make use of the Eikonal equation and the Prim's algorithm to form the initial curve, which may contain sharp corners and intersections. We also introduce a more powerful curve smoothing method. Our generalized FCT algorithm is able to trace a wide range of complicated curves, such as handwritten Chinese characters.