Ternary Blending Operations

We discuss new analytical formulations for localized and controllable blending operations in the functionbased solid modeling. The blending set operations are defined using R-functions and displacement functions with the localized area of influence. The shape and location of the blend are controlled by an additional bounding solid thus turning the operation into a ternary one. We also describe a new approach to solving the problem of shape metamorphosis between k-dimensional shapes by applying space-time bounded blending to the specially constructed (k+1)dimensional half-cylinders and making cross-sections for getting intermediate shapes under the transformation. 1 Blending in solid modeling Blending operations in solid modeling generate smooth transitions between two or several surfaces. Blending is also considered a natural property of implicit surfaces, where the basic operation is an algebraic sum (or difference) between skeleton-based scalar fields. Blending operations are typically used in computer-aided design for modeling fillets and chamfers. These operations are usually smooth versions of set-theoretic operations on solids (intersection, union, and difference), which approximate exact results of these operations by rounding sharp edges and vertices. The major requirements to blending operations [1] are tangency of the blend surface with the initial surfaces, automatic clipping of unwanted parts of the blending surface, C1 continuity of the blending function everywhere in the domain, support of added and subtracted material blends. Special attention is paid to the intuitive control of the blend shape and position: the construction of the blend and its parameters should have clear geometric interpretation