The Visualization of Second-Order Tensor Fields

<p>The field of scientific visualization covers the study of visual representations of sci- entific data. These data often consist of continuous scalar vector or second-order tensor fields extending across a n-dimensional space with n less than 1. While research efforts aimed at developing efficient representations of scalar and vector data have led to important achievements this dissertation is the first extensive study of visualization techniques for second-order tensor fields. Tensor fields are multivariate; they can embed as much information as n squared scalar fields or equivalently n vector fields. Visualizing continuous tensor data is therefore difficult mainly because the underlying continuity must be rendered while visual clutter has to be avoided. We develop a theoretical ground work for the visualization of symmetric tensor fields by studying their geometry and their topological structure and by designing at each step appropriate icons to represent the information. We also extend some of our concepts to asymmetric tensor data. First we design icons that emphasize the continuity of the tensor data overcoming some of the limitations of discrete point icons. A n-dimensional symmetric tensor field is equivalent to n orthogonal families of smooth and continuous curves that are tangent to the eigenvector fields. For n = 2 we generate textures to render these trajectories and for n = 3 we use numerical integration. To fully represent the tensor data we surround the resulting trajectories by tubular surfaces that represent the transverse eigenvectors-we call these surfaces hyper-streamlines. We also define the concept of a solenoidal tensor field and we show that its hyperstreamlines possess geometric properties similar to the streamlines of solenoidal vector fields. Then we analyze the topology of symmetric tensor fields by using a formalism which is analogous to the phase-space analysis of dynamical systems.</p><p>This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact and remains as true to the original work as possible. Therefore you will see the original copyright references library stamps (as most of these works have been housed in our most important libraries around the world) and other notations in the work.</p><p>This work is in the public domain in the United States of America and possibly other nations. Within the United States you may freely copy and distribute this work as no entity (individual or corporate) has a copyright on the body of the work.</p><p>As a reproduction of a historical artifact this work may contain missing or blurred pages poor pictures errant marks etc. Scholars believe and we concur that this work is important enough to be preserved reproduced and made generally available to the public. We appreciate your support of the preservation process and thank you for being an important part of keeping this knowledge alive and relevant.</p><br>