Galina Pasko

Bounded blending for function-based shape modeling

New analytical formulations of bounded blending operations can enhance function-based constructive shape modeling. In this article, we introduce bounded-blending operations that we define using R-functions and displacement functions with the localized area of influence. We define the shape and location of the blend by control points on the surfaces of two solids or by an additional bounding solid. We can apply our proposed blending using a bounding solid to a single selected edge or vertex. We also introduce new multiple blends and a partial edge blend. Our description supports set-theoretic operations on blends and blends on blends - that is, recursive blends. In this sense, our proposed operations could replace pure set-theoretic operations in the construction of a solid without rebuilding the entire construction tree data structure. Our proposed blending method can have application in interactive design.

Bounded blending operations

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.

Cultural Heritage Preservation Using Constructive Shape Modeling

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.

Space–time blending

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

Rendering trimmed implicit surfaces and curves

Models of implicit surfaces and curves trimmed by a solid are discussed in the context of dimensionally heterogeneous object modeling. Both a carrier surface and a trimming solid are modeled using the function representation. Algorithms for polygonization of trimmed surfaces and curves, as well as ray-tracing of trimmed surfaces are described. Illustrative and CAD related examples are given.

Trimming implicit surfaces

Algorithms for trimming implicit surfaces yielding surface sheets and stripes are presented. These two-dimensional manifolds with boundaries result from set-theoretic operations on an implicit surface and a solid or another implicit surface. The algorithms generate adaptive polygonal approximation of the trimmed surfaces by extending our original implicit surface polygonization algorithm. The presented applications include modeling several spiral shaped surface sheets and stripes (based on M. Escher’s artworks) and extraction of ridges on implicit surfaces. Another promising application of the presented algorithms is modeling heterogeneous objects as implicit complexes.

Digitally Interpreting Traditional Folk Crafts

Preserving cultural heritage requires that objects persist throughout time and continue to communicate an intended meaning. Owing to the decreasing number of masters of folk crafts, fading technologies, and crafts losing economic ground, computer-based preservation and interpretation of such crafts is necessary. To fabricate and preserve traditional crafts, a long-term applied-research project has combined mathematics and software tools with compact, cheap, and environmentally friendly desktop fabrication tools, including 3D printers. Case studies involving the digital capture of Japanese lacquerware and Norwegian carvings illustrate the projects modeling approach and fabrication system. Besides modeling existing artifacts, the project includes Web presentations of the models, automated model fabrication, and experimental manufacturing of new designs and forms.

Real-time space-time blending with improved user control

In contrast to existing methods of metamorphosis based on 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 has 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 propose several techniques for more intuitive user control for space-time blending. 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. This gives more control to the user. The approach is further extended with the introduction of an additional non-linear deformation operation to the pure space-time blending. The proposed techniques have been implemented and tested within an industrial computer animation system. Moreover, this method can now be employed in real-time applications taking advantage of modern GPUs.

SMI 2011: Full Paper: Feature based volumes for implicit intersections

The automatic generation of volumes bounding the intersection of two implicit surfaces (isosurfaces of real functions of 3D point coordinates) or feature based volumes (FBV) is presented. Such FBVs are defined by constructive operations, function normalization and offsetting. By applying various offset operations to the intersection of two surfaces, we can obtain variations in the shape of an FBV. The resulting volume can be used as a boundary for blending operations applied to two corresponding volumes, and also for visualization of feature curves and modeling of surface based structures including microstructures.

Space-time modeling and analysis

In this survey, the problem of general type shape metamorphosis is considered as a typical space-time modeling operation. A new approach based on bounded blending of space-time half-cylinders is described. Detection and classification of critical points of shape topological changes on the time axis are presented. Examples in 2D and 3D spaces are given.