Results

So far, we experimented with four data sets to test out the proposed degenerate tensor extraction algorithm. The first is a 2D rectangular patch with symmetric 3D tensors at the four corners that have been set randomly. The tensor values within the patch are obtained through linear interpolation. This synthetic data corresponds to tensors on a face of a 3D cell. The second is a 3D cell with symmetric 3D tensors on its eight corners which are also set randomly. The third is the stress tensor data in a semi-infinite volume with two point loads. The last is the deformation tensors in a computed flow past a cylinder with a hemisphere cap. In all the figures, the colors are mapped to the tensor discriminant with blue mapped to zero and warmer colors to higher values. Degenerate tensors can be found in the blue regions.

2D randomized data:

(a) (b) (c) (d)

Figure 1: White dots are degenerate points indicating places where all seven constraint functions are zero. Each colored curve corresponds to a constraint function being equal to zero. Places where multiple curves intersect are where multiple constraint functions are satisfied simultaneously. The background is pseudo-colored by the discriminant functions. The data is a 2D slice of a randomly generated 3D tensor field.
Higher Order Degeneracy:

(a) Linear Interpolation (b) Tricubic Spline Interpolation

Figure 2: Closeup of the bifurcating feature lines below the point loads in the double point load data set. Feature lines are colored by the eigen difference while the volume is colored by the discriminant using cool colors for low discriminant values (closer to feature lines) and warm transparent colors for distant values.
3D randomized data:

Figure 3: Randomly generated 3D tensors. Warmer line colors are closer to type A degenerate points where major and medium hyperstreamlines intersect, while cooler line colors are closer to type B degenerate points where medium and minor hyperstreamlines intersect. The rest of the volume is pseudo-colored by the discriminant using cool colors for low discriminant values (closer to feature lines) and warm transparent colors for distant values.
Double point load stress tensors:

(a) Oblique view (b) Front view (c) Side view (d) Top view

(e) Unsymmetric loads

Figure 4: Double point load data. Yellow arrows indicate point load, while the 2 magenta spheres show the location of the triple degenerate points. Feature lines are colored by the eigen difference while the volume is colored by the discriminant.
Flow past a cylinder with a hemisphere cap:

Figure 5: Degenerate lines in deformation tensors of flow past a cylinder with a hemispherical cap. Feature lines are colored by the eigen difference.
Stress tensors with two opposite point loads.

Figure 6: Degenerate lines in Stress tensors with two opposite point loads. Feature lines are colored by the eigen difference. The data set is courtesy of Boris Jeremic.

Animations:

Use Windows Media Player to view the animation of feature lines on a stress tensor field with a tensile and a compressive point load. The data set is courtesy of Boris Jeremic:

Both type-P and type-L feature lines: (download)

Only type-P lines: (download)

Only type-L lines: (download)


(a) Oblique view	(b) Front view	(c) Side view	(d) Top view

(e) Unsymmetric loads


(a) Linear Interpolation	(b) Tricubic Spline Interpolation

Results

2D randomized data:

Higher Order Degeneracy:

3D randomized data:

Double point load stress tensors:

Flow past a cylinder with a hemisphere cap:

Stress tensors with two opposite point loads.

Animations:

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