Blending Implicit Shapes Using Fuzzy Set Operations

Implicit modelling is a powerful technique to design geometric shapes, where a geometric object is described by a real function. In general, the real functions used in implicit modelling are unbounded and can take any values in space R. In general, the shapes described by different level sets of an unbounded implicit function can be varied significantly and are very unpredictable. In addition, allowing the underlying implicit function to take negative values also makes the construction of shape blending operations a difficult task. In this paper, we propose an implicit shape modelling technique, where each implicit shape is represented as the membership function of a fuzzy set which is bounded and nonnegative with value taken in [0, 1]. The most obvious benefit of representing an implicit shape as fuzzy set is that the blending of a set of implicit shapes is simply a problem of aggregating a set of fuzzy sets, which can be done in various ways by choosing a proper fuzzy set aggregation operator from a wide variety of fuzzy set operations. Key–Words: Implicit curves and surfaces, isosurfaces, blending operations, Generalized algebraic operations, Piecewise algebraic operations, Fuzzy sets, Soft computing