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New analytical formulations of bounded blending for functionally defined set-theoretic operations are proposed. 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 is defined by control points on the surfaces of two solids or by an additional bounding solid. The proposed blending using a bounding solid can be applied to a single selected edge or a vertex. We introduce new types of blends such as a multiple blend with the disconnected bounding solid and a partial edge blend. It is shown to have versatile applications in interactive design.

Issues of digital preservation of shapes and internal structures of historical cultural objects are discussed. An overview of existing approaches to digital preservation related to shape modeling is presented and corresponding problems are considered. We propose a new digital preservation paradigm based on both constructive modeling reflecting the logical structure of the objects and open standards and procedures. Constructive Solid Geometry (CSG) and Function Representation (FRep) are examined and practically applied as mathematical representationsproducing compressed yet precise data structures, thus providing inter-operability between current and future computer platforms crucial to archiving. Examples of CSG reconstruction of historical temples and FRep modeling of traditional lacquer ware are given. We examine the application of fitting of a parameterized FRep model to a cloud of data points as a step towards automation of the modeling process. Virtual venues for public access to cultural heritage objects including real time interactive simulation of cultural heritage sites over the Web are discussed and illustrated.

Shape transformation between objects of different topology and positions in space is an open modelling problem. We propose a new approach to solving this problem for two given 2D or 3D shapes. The key steps of the proposed algorithm are: increase dimension by converting two input kD shapes into half-cylinders in (k þ 1)D space-time, applying bounded blending with added material to the half-cylinders, and making cross-sections for getting intermediate shapes under the transformation. The additional dimension is considered as a time coordinate for making animation. We use the bounded blending set operations in spacetime defined using R-functions and displacement functions with the localized area of influence applied to the functionally defined half-cylinders. The proposed approach is general enough to handle input shapes with arbitrary topology defined as polygonal objects with holes and disjoint components, set-theoretic objects, or analytically defined implicit surfaces. The obtained unusual amoeba-like behaviour of the shape combines metamorphosis with the non-linear motion.

A well-known definition of visualization is the mapping of initial data to a visual representation, which can be perceived and interpreted by humans. Human senses include not only vision, but also hearing, sense of touch, smell and others including their combinations. Visual analytics and its more general version that we call Multisensory Analytics are areas that consider visualization as one of its components. We present a particular case of the multisensory analytics with a hybrid visual-auditory representation of data to show how auditory display can be used in the context of data analysis. Some generalizations based on using real-valued vector functions for solving data analysis problems by means of multisensory analytics are proposed. These generalizations might be considered as a first step to formalization of the correspondence between the initial data and various sensory stimuli. An illustration of our approach with a case study of analysis of a scalar field using both visual and auditory data representations is given. 322Malikova E.