Most of the existing methods of metamorphosis are based on the interpolation schemes, space-time blending is a geometric operation of bounded blending performed in the higher-dimensional space. It provides transformations between shapes of different topology without necessarily establishing their alignment or correspondence. The original formulation of space-time blending had several problems: fast uncontrolled transition between shapes within the given time interval, generation of disconnected components, and lack of intuitive user control over the transformation process. We improve the original technique to provide the user with a set of more intuitive controls. The problem of the fast transition between the shapes is solved by the introduction of additional controllable affine transformations applied to initial objects in space-time. The approach is further extended with the introduction of an additional non-linear deformation operation. The proposed techniques have been implemented and tested within an industrial computer animation system. We have also implemented our method on the GPU, so that it can be employed in real-time applications.

We further develop a new approach to shape metamorphosis using bounded blending operations in space-time. The key steps of the metamorphosis algorithm are: dimension increase by converting two input kD shapes into half-cylinders in (k+1)D space-time, applying bounded blending union with added material to the half-cylinders, and making cross-sections for getting intermediate shapes under the transformation. This approach is extended here in two directions. First, the problem of "jump" in animation or the rapid transition between shapes in the given interval is solved using "smoothed" versions of half-cylinders which undergo bounded blending. Second, the approach is extended to 3D initial and final shapes with the bounded blending union operation applied to the corresponding "smoothed" 4D space-time half-cylinders.

Abstract 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 representations producing 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.

Abstract 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+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 space–time 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. Copyright © 2004 John Wiley & Sons, Ltd.

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.