Shapes in Vector Fields

Abstract

Geometric shapes are the basic building blocks of any graphicsrelated application.The effective manipulation of shapes is therefore of central interest formany relevant problems.In particular, there is a growing demand for high-quality nonlineardeformations for shape modeling and animation.The application of vector fields that guide a continuous deformation is apractical approach for their computation.It turns out that typically challenging nonlinear problems can be solved inan elegant way using such vector field-based methodologies.This thesis presents novel approaches and prospects for vector field-basedmanipulation of geometric shapes (Part I).Thereafter, also the definition of geometric shapes by means ofvector fields is examined (Part II).Depending on the specific shape representation and the concrete modelingproblem, different types of vector fields are required:a family of generalized vector field energies is introduced that enablesnear-isometric, near-conformal, as well as near-authalic continuousdeformations of planar and volumetric shapes.It is demonstrated how near-isometric surface and volume-preservingisosurface 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, aconceptual simple but effective generalized energy regularization isproposed, which is not only beneficial for continuous deformations butadditionally enhances a variety of related geometry processing methods.In the second part of the thesis vector fields are not considered torepresent deformations anymore.Instead, they are interpreted as flow fields that define characteristicshapes such as stream surfaces:a deformation-based approach for interactive flow exploration and theextraction of flow-tangential and flow-orthogonal surfaces is proposed.It is shown how an unified computational framework yields parametrizationsthat are particularly useful for surface-based flow illustrations.Finally, an automatic method for the selection of relevant stream surfacesin complex flow data sets is presented that is based on a new surface-basedintrinsic quality measure.The usefulness of the newly developed methods is shown by applying them to anumber of different geometry processing and visualization problems.