Fuzzy interpolation offers the potential to model problems with sparse rule bases, as opposed to dense rule bases deployed in traditional fuzzy systems. It thus supports the simplification of complex fuzzy models and facilitates inferences when only limited knowledge is available. This paper first introduces the general concept of representative values (RVs), and then uses it to present an interpolative reasoning method which can be used to interpolate fuzzy rules involving arbitrary polygonal fuzzy sets, by means of scale and move transformations. Various interpolation results over different RV implementations are illustrated to show the flexibility and diversity of this method. A realistic application shows that the interpolation-based inference can outperform the conventional inferences