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Minor thesis of Martin ManzerInteractive shape deformation on triangular meshes based on differential coordinatesChair of Computer Graphics and Visualization
MotivationBeside modeling operations such as morphing and blending, shape deformation has been a subject of increasing attention in recent years. Depending on what modeling metaphor is used, shape deformation can be understood as the displacement of a manipulation tool followed by a redefinition of the model geometry, whereas the topology of the mesh is not changed. The design of modeling systems is still a challenging problem, since they need to satisfy a number of crucial requirements. They should be intuitive and easy to use with minimal user interaction. This implies robustness and efficiency to provide the user with an interactive response. In contrast to other techniques, the usage of differential coordinates is a well suited method to meet these requirements. DescriptionThe work introduces the motivation and definition of differential coordinates. In contrast to absolute coordinates, which can only tell the spatial location of each vertex, they capture local shape properties of the surface and strive to preserve them while the user deforms the mesh. The main characteristics and problems that arise in the deformation using differential coordinates are explained with three examples: Pyramid coordinates, Laplacian coordinates and differential coordinates in 2D. All methods are extensively illustrated and compared. Furthermore, we illuminate implementation aspects. Especially, the application of sparse matrices is of major importance. ResultsDifferential coordinates supply a very intuitive modeling metaphor by simply selecting and dragging parts of a mesh. The user is able to control the influence area by choosing a region of interest. The deformation is done at interactive rates and generates pleasing deformed surfaces that imitate the rigidity of objects such that the result satisfies what the user actually intended to do. Nevertheless, the shape preservation property of differential coordinates may also produce unwanted results. We believe that for models with an obvious jointed structure skeleton-techniques are the preferable method, since they preserve the intrinsic physical structure.
Future workTo obtain the actual deformed geometry from the differential representation we need to apply a global reconstruction. This points out to be the most crucial operation in interactive editing. The necessary calculations must happen in real-time. They highly depend on the chosen local differential representation of the surface. Further considerations would possibly remedy the problem. Furthermore, it is known that deformation based on differential coordinates suffers from a non-rigid behavior under large deformation. This is because volume preservation is not handled explicitly. Only some recent articles address this problem. We believe that an adaptation of the 2D example to 3D is worthwhile. Download |
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