Sampling methodologies are widely used in network measurements and other related fields. Most applications mainly focus on parent population statistical metrics estimation of interest. Recent researches reveal that many aspects of network characters present heavy-tailed distribution or self-similarity. These properties might cause a heavy passive effect on the estimation accuracy. In other circumstances, there exist demands on modeling the characteristics of a network in network operation. To develop an accurate model for network character is much difficult. From a broader view, these applications are treated as special cases of fitting problems of planar data set or time series in applied mathematics. In the paper, a Fitting-based adaptive sampling methodology (FASM) is developed for reconstructing the evolution of some network characteristics (model). The contributions of the paper include: (1) Adopting a Piecewise Linear Function Approximation scheme to provide a more accurate approximation of the true character. (2) The statistical metric derived from the FASM provides a much more stable and accurate estimation than other popular methodologies under the same sampling size. Experiments based on two measurement traces show that the FASM can dramatically reduce the number of samples while retaining the same approximating residual error as others. (3) The variance of sampling size is more stable than those of other probability sampling schemes.