Hyper streamline

With Hyper streamlines ( English singular Hyper Streamline ) can be symmetric, real tensor second stage with non-negative eigenvalues depict analogous to power lines of a vector field. They were described in 1993 by Lambertus Hesselink and Thierry Delmarcelle.

Such a representation of the tensor field by hypercurrent lines is particularly useful if one of the eigenvectors is related to a particle flow.

Action

A tensor field maps every point in space to a tensor. In order to be able to imagine this arrangement, a visualization is helpful.

A symmetric second order tensor is given by a square, symmetric matrix . The essential information is not contained in the entries of the matrix, but its eigenvectors and eigenvalues. According to the spectral theorem , the eigenvectors of symmetrical matrices are perpendicular to one another. The tensor can thus be represented by three mutually perpendicular vectors whose lengths are precisely the eigenvalues. Negative eigenvalues ​​would already lead to a representation problem at this point, which is why one restricts oneself to symmetrical tensor fields with non-negative eigenvalues.

One possible visualization of the tensor field is to either display these three vectors as arrows at certain points in space (e.g. a grid), or to create an ellipsoid using these arrows , with the semiaxes of the spanning ellipse in the direction of the eigenvectors and as length are given by the eigenvalues.

Hypercurrent lines, on the other hand, do not represent the tensor field at grid points, but rather through tubes that smear the ellipsoids described above along the direction of the first eigenvector (e.g. the one with the greatest eigenvalue). The center line of the hose is thus precisely the streamline that is obtained from the vector field of the respective "first" eigenvector. The cross-section of the hypercurrent line is elliptical, the semiaxes of the ellipse being given by the directions of the other two eigenvectors and their eigenvalues. Since the information about the first eigenvalue was lost, this length is coded using different colors along the tube.

example

Example of an eigenvalue equation with a symmetric matrix ,

${\displaystyle {\begin{pmatrix}2&0&-1\\0&2&-1\\-1&-1&3\end{pmatrix}}{\begin{pmatrix}a\\b\\c\end{pmatrix}}=\lambda {\begin{pmatrix}a\\b\\c\end{pmatrix}}\,,}$

${\displaystyle \lambda =1:\,\mathbf {e} _{1}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1\\1\\1\end{pmatrix}}\,,\ \lambda =2:\,\mathbf {e} _{2}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-1\\0\end{pmatrix}}\,,\ \lambda =4:\,\mathbf {e} _{3}={\frac {1}{\sqrt {6}}}{\begin{pmatrix}1\\1\\-2\end{pmatrix}}\,.}$

Web links

• Heike Jänicke, Visualization I, 8 , lecture notes Uni Heidelberg (PDF; 4.2 MB)
• Burkhard Wünsche, Hyperstreamlines , IEEE Computer Graphics and Applications, 13 (1993) 25-33