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, C 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. ∗IT Institute, Kanazawa Institute of Technology, [email protected] †IT Institute, Kanazawa Institute of Technology and Hosei University, [email protected] ‡IT Institute, Kanazawa Institute of Technology,