Geometric shapes are the basic building blocks of any graphics related application. The effective manipulation of shapes is therefore of central interest for many relevant problems. In particular, there is a growing demand for high-quality nonlinear deformations for shape modeling and animation. The application of vector fields that guide a continuous deformation is a practical approach for their computation. It turns out that typically challenging nonlinear problems can be solved in an elegant way using such vector field-based methodologies. This thesis presents novel approaches and prospects for vector field-based manipulation of geometric shapes (Part I). Thereafter, also the definition of geometric shapes by means of vector fields is examined (Part II).
Depending on the specific shape representation and the concrete modeling problem, different types of vector fields are required: a family of generalized vector field energies is introduced that enables near-isometric, near-conformal, as well as near-authalic continuous deformations of planar and volumetric shapes. It is demonstrated how near-isometric surface and volume-preserving isosurface deformations are computed by a similar framework. Furthermore, an integration-based pose correction method is presented. Based on a generic energy description that incorporates energy smoothness, a conceptual simple but effective generalized energy regularization is proposed, which is not only beneficial for continuous deformations but additionally enhances a variety of related geometry processing methods.
In the second part of the thesis vector fields are not considered to represent deformations anymore. Instead, they are interpreted as flow fields that define characteristic shapes such as stream surfaces: a deformation-based approach for interactive flow exploration and the extraction of flow-tangential and flow-orthogonal surfaces is proposed. It is shown how an unified computational framework yields parametrizations that are particularly useful for surface-based flow illustrations. Finally, an automatic method for the selection of relevant stream surfaces in complex flow data sets is presented that is based on a new surface-based intrinsic quality measure. The usefulness of the newly developed methods is shown by applying them to a number of different geometry processing and visualization problems.


  title = {Shapes in Vector Fields},
  author = { {Martinez~Esturo}, J.},
  school = {University of Magdeburg},
  year = {2013}


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