A method for approximating contour figures with piecewise-linear polygons using a criterion function based on regularization theory is presented. In this method, it is possible to select the degree of fineness of the approximation to a given contour figure with a parameter in the criterion function, which represents a tradeoff between the fitness and the simplicity of the polygonal model. However, it is shown that, for a given figure, only a few types of stable models optimize the criterion with a variety of values of the parameter. This means that only a few types of polygons may be suitable approximations for representing the original characteristic features of the figure. This method derives such polygons automatically from the given figures. Thus, this approach shows potential as a powerful method for the analysis and interpretation of planar contour figures.