If the number of variables is growing the size of fuzzy rule bases increase exponentially. To reduce size and inference/control time, it is often necessary to deal with sparse rule bases. In such bases, classic algorithms such as the CRI of Zadeh and the Mamdani-method do not function. In such rule bases, rule interpolation techniques are necessary. The linear rule interpolation (KHinterpolation) based on the Fundamental Equation of Interpolation introduced by Koczy and Hirota is suitable for dealing with sparse bases, but this method often results in conclusions which are not directly interpretable, and need some further transformations. One of the possible ways to avoid this problem is the interpolation method based on the conservation of fuzziness, proposed recently by Gedeon and Koczy for trapezoidal fuzzy sets. In this paper, a refined version of that method will be presented that is fully in accordance with the Fundamental Equation, with extensions to multiple dimensions, and then to arbitrarily shaped membership functions. Several possibilities for the latter will be shown.